Y-1

Y.0 EXAMPLE CALCULATIONS—INITIAL CORRELATION TEST

This section presents several examples of initial correlation test calculations. For each of the

examples, a brief description of the correlation test scenario is provided, followed by a summary

and discussion of the results of the calculations. For the first example, all of the equations used

to evaluate the data and some sample calculations are presented. The calculations for all of the

examples were performed using the Correlation Test Calculation Spreadsheet, which is described

in Section X. Appendix A includes the entire spreadsheet output for each of the examples.

Y.1 EXAMPLE 1: HOW DO I DEVELOP AN INITIAL CORRELATION?

Y.1.1 Correlation Test Scenario

A correlation test is performed on a boiler at Facility A. The boiler is equipped with an in situ

PM CEMS that responds to PM concentrations at actual conditions. The emission limit, when

converted to units that are consistent with the PM CEMS sampling conditions, is 20.0 mg/acm. 

Twenty-four test runs are performed using a single Method 5 sampling train. The facility

determines that four of the test runs (Runs 7, 8, 9, and 18) were suspect because of problems

with the sampling train. Table Y-1 summarizes the correlation test data. Figure Y-1 shows a

plot of the data.

Table Y-1. Summary of Correlation Test Data for Facility A

Run no.

PM CEMS response,

mA

PM concentration,

mg/acm

1 4.90 2.1

2 7.60 2.9

3 7.80 5.1

4 9.20 4.8

5 10.10 2.9

6 11.06 6.3

7 13.40 0.2

8 16.10 0.6

9 15.80 0.3

10 12.00 8.4

(continued)
Table Y-1. (continued)

Run no.

PM CEMS response,

mA

PM concentration,

mg/acm

Y-2

0

2

4

6

8

10

12

14

16

18

20

0 5 10 15 20 25 30 35

PM CEMS Response, mA

PM Concentration, mg/acm

Figure Y-1. Plot of the correlation test data for Facility A.

11 18.20 8.4

12 15.70 4.9

13 13.90 6.3

14 19.70 13.0

15 21.02 9.1

16 22.30 13.9

17 24.10 8.7

18 25.90 0.6

19 24.70 12.6

20 26.40 15.9

21 29.08 16.2

22 29.30 13.5

23 30.50 15.9

24 33.10 18.5
Y-3

Must I use all of the test data to develop the correlation?

Section 8.6(3) of PS-11 requires a minimum of 15 valid test runs for developing the correlation,

and Section 8.6(3)(iii) allows you to discard up to 5 test runs without explanation. Additional

test runs can be discarded if the basis for rejecting the runs is specified in the manual test

method, in PS-11, or in the quality assurance plan for the source. In this example, an additional

nine test runs were performed, so it is not necessary to use all of the test data. Five of the runs

can be discarded without explanation. As indicated in Figure Y-1, the results from test Runs 7,

8, 9, and 18 appear to be outliers. If the four outliers are discarded, there are 20 runs remaining,

which satisfies the minimum requirement of 15 runs for developing the initial correlation. Note

that Section 8.6(3)(ii) of PS-11 still requires you to report all of the data, even for those runs that

are rejected.

How do I develop a linear correlation curve from the data?

To develop a linear correlation model, follow the procedures described in Section 12.3(1) of

PS-11. The form of the linear model is provided in Equation 11-3 of PS-11:

y b bx $ = + 0 1

where

í = the predicted PM concentration

b0 = the intercept for the correlation curve

b1 = the slope of the correlation curve

x = the PM CEMS response value.

To begin, the mean PM CEMS response (xG) and mean PM concentration (yG) are calculated as

follows:

x

n

x ( . . . .... . ) . i

i

n

= = ∑ + + ++ = =

1 1

20 4 9 7 6 7 8 331 1853

1

y

n

y ( . . . ....+ . ) . i

i

n

= = + ++ = ∑

=

1 1

20 21 2 9 51 185 9 47

1

The variables Sxx and Sxy, which are presented in Equation 11-7 of PS-11, are calculated next as

follows:

( ) (( ) ( ) ( ) ) S xx xx i

i

n

= − = − + − ++ − =

=

∑ 2 22 2

1

4 9 1853 7 6 1853 331 1853 1 442 2 . . . . . . ,. K

( )( ) ( )( ) ( )( )

( )( )

S x xy y xy i i

i

n

= − − = − − + − − ++

− −=

=

∑ 4 9 18 53 2 1 9 47 7 6 18 53 2 9 9 47

331 1853 185 9 47 7852

1

. . .. . . . .

. . .. .

K
Y-4

The slope of the correlation curve (b1) is determined by Equation 11-6 of PS-11:

b

S

S

xy

xx

1

7852

1 442 2 == = 0544 ., . .

The intercept of the correlation curve is determined using Equation 11-4 of PS-11:

b ybx 0 1 =−⋅ = 9 47 0544 1853 0 621 . (. . ) . − × = −

The resulting equation for the correlation curve is as follows:

í = !0.621 + 0.544x

How do I determine the linear correlation coefficient?

The linear correlation coefficient (r) is calculated using the parameters SL and Sy. The parameter

SL is defined in Equation 11-9 as follows:

S ( ) ny y L ii

i

n

= − −

=

∑ 1

2

2

1

$

To calculate SL, the linear correlation equation must be used to determine the predicted PM

concentration (í) for each test run. For example, for Run 3, the PM CEMS response is 7.8 and

the predicted PM concentration is

y$ =− + = 0 621 0544 7 8 3626 . . .. ( )

Table Y-2 shows the results of the calculations of í. Using these values, the calculation for SL is

as follows:

(( ) ( ) ( ) ( ) ) S L = − − + − + − ++ − =

1

20 2

2 047 21 3517 2 9 3626 51 17 401 185 1910 2 22 2 .... .. . . . K

Table Y-2. Predicted PM Concentrations (í) for Linear Correlation

Run no.

PM CEMS 

response, mA

PM concentration,

 mg/acm

Predicted PM

concentration (í)

a

1 4.9 2.1 2.047

2 7.6 2.9 3.517

3 7.8 5.1 3.626

4 9.2 4.8 4.388

5 10.1 2.9 4.878

(continued)
Table Y-2. (continued)

Run no.

PM CEMS 

response, mA

PM concentration,

 mg/acm

Predicted PM

concentration (í)

a

Y-5

6 11.06 6.3 5.401

10 12 8.4 5.913

11 18.2 8.4 9.289

12 15.7 4.9 7.928

13 13.9 6.3 6.947

14 19.7 13 10.105

15 21.02 9.1 10.824

16 22.3 13.9 11.521

17 24.1 8.7 12.501

19 24.7 12.6 12.828

20 26.4 15.9 13.753

21 29.08 16.2 15.213

22 29.3 13.5 15.332

23 30.5 15.9 15.986

24 33.1 18.5 17.401

a

 Calculated using the correlation equation: í = -0.621 + 0.544x.

The parameter Sy is calculated using Equation 11-15 of PS-11 as follows:

( ) ( )( )( ) ( ) S

y y

n y

i

i

n

=

−

− = − + − + − ++ −− = =

∑ 2

1

2 22 2

1

21 9 47 2 9 9 47 51 9 47 185 9 47

20 1 5095 .. . . .. .. . K

With the values for SL and Sy, the correlation coefficient is calculated using Equation 11-14 as

follows:

( )

( ) r

S

S

L

y

=− =− = 1 1

1910

5095

0 927

2

2

2

2

.

. .

The correlation coefficient (0.927) is greater than 0.85 and, therefore, satisfies PS-11.

Note: For data sets that show little to no correlation, the value for SL can be greater than Sy. In

such cases, the correlation coefficient (r) calculated using Equation 11-14 is undefined because it

entails determining the square root of a negative number. Even though the correlation

coefficient cannot be determined, it can be assumed that the value for r does not meet the

criterion of 0.85 specified by PS-11.
Y-6

How do I calculate the confidence interval half range percentage for the linear correlation?

The half range for the 95 percent confidence interval (CI) for the predicted PM concentration at

the mean value of x is determined using Equation 11-8 of PS-11:

CI t S

n df, a L = ⋅− 1 2 −

1

The variable tdf, 1-a/2 is obtained from a table of t-statistic values. The degrees of freedom (df)

equals n-2, and the value for the t statistic for this example is 2.101. With this value and the

value for SL calculated above, CI is calculated as follows:

CI = = 2101 191 ( ) 1

20 .. . 0897

The emission limit in this example is 20.0 mg/acm. To determine CI as a percentage of the

emission limit, Equation 11-10 is used as follows:

CI

CI

EL

% .

. =× = × = 100% . 089720 0 100% 4 49%

The resulting value for CI% is 4.49 percent. Because this value is less than 10 percent, the

correlation satisfies the criterion for confidence interval half range.

How do I calculate the tolerance interval half range percentage for the linear correlation?

Equation 11-11 of PS-11 is used to calculated the tolerance interval half range (TI) at the mean

value of x:

TI k S T L = ⋅

The parameter kT is determined using Equation 11-12 of PS-11:

k uv T n df = ⋅ ′

The parameters unN and vdf are determined from Table 1 of PS-11. In this example, nN is the

number of test runs used for the correlation (20), unN

 = 1.177, and vdf equals 1.384 for n-2 degrees

of freedom. With these values, kT and TI are calculated as follows:

k ( ) T u

n

v

df = . . . ′ ⋅= = 1177 1384 1629

TI k S ( ) T L =⋅= = 1629 191 3111 .. .

With an emission limit of 20.0 mg/acm, the tolerance interval as a percentage of the emission

limit is determined using Equation 11-13 as follows:
Y-7

TI

TI

EL

% .

. =× = × = 100% . 311120 0 100% 156%

The resulting value for TI% is 15.6 percent. Because this value is less than 25 percent, the

correlation satisfies the criterion for tolerance interval half range. In summary, the linear

correlation developed from the data satisfies all of the criteria specified in PS-11.

How do I develop a polynomial correlation curve from the data?

A polynomial correlation curve can be expressed by the following:

y b bx bx $ =+ + 01 2

2

The procedures for developing a polynomial correlation curve are described in Section 12.3(2) of

PS-11. The first step is to determine the values for the parameters S1 through S7, as defined in

Equations 11-17 and 11-18 of PS-11. These calculations are summarized as follows:

S x( ) ( ) i

i

n

1

1

= = + + ++ = 4 9 7 6 7 8 331 370 7 =

∑ ... . . K

S x( ) ( ) i

i

n

2

2 222 2

1

= = + + ++ = 4 9 7 6 7 8 331 8 312 =

∑ ... . , K

S x( ) ( ) i

i

n

3

3

1

333 3 = = + + ++ = 4 9 7 6 7 8 331 208 377 =

∑ ... . , K

S x( ) ( ) i

i

n

4

4 444 4

1

= = + + ++ + 4 9 7 6 7 8 331 5 574 977 =

∑ . . . . ,, K

S y( ) ( ) i

i

n

5

1

= = + + ++ = 21 2 9 51 185 189 4 =

∑ ... . . K

S xy ( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) i i

i

n

6

1

= = + + ++ = 4 9 21 7 6 2 9 7 8 51 331 185 4 295 =

∑ . . .. .. . . , K

S xy ( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) i i

i

n

7

2 22 2 2

1

= = + + ++ = 4 9 2 1 7 6 2 0 7 8 51 331 185 108 567 =

∑ . . .. .. . . , K

Next, the parameter det A must be calculated using Equation 11-22 of PS-11. For this example,

the calculation is as follows:

det A nS S S S S S S S S S n S S S S S S =⋅ ⋅ − ⋅ ⋅ + ⋅ ⋅ − ⋅ ⋅ + ⋅ ⋅ − ⋅⋅ = 2 4 2 2 2 132 33 213 411

( ) 20 8 312 5 574 977 8 312 8 312 8 312 370 7 208 377 8 312 208 377 208 377 20 ( )( ) ( )( ) , ,, , , , . , , , , −+ − + ( ) ( )( ) ( )( )( )

( ) 8 312 370 7 208 377 5 574 977 370 7 370 7 2120 10 ( )( )( )( )( ) 9 , . , ,, . . . − =×

The values for the correlation curve parameters b0, b1, and b2 can now be determined using

Equations 11-19 to 11-21 of PS-11:
Y-8

( ) b

SSS SSS SSS SSS SSS SSS

A 0

524 137 263 722 335 461

= ⋅⋅ +⋅⋅ + ⋅⋅− ⋅⋅ − ⋅⋅− ⋅⋅

det

= ( )( )( ) ( )( )( ) ( )( )( ) ×++ −

1

212 109 189 8 312 5 574 977 370 7 208 377 108 567 208 377 4 295 208 377 . [ , ,, . , , , , ,

( )( )( ) ( ) 108 567 8 312 8 312 208 377 208 377 189 5 574 977 4 295 370 7 , , , , , , , , .] − − ( )( ) ( )( )( )

= 0 420 .

( ) b

nS S S S S S S S S S S S S n S S S

A 1

64 532 217 262 73 415

= ⋅ ⋅ + ⋅ ⋅ + ⋅ ⋅ − ⋅ ⋅ − ⋅ ⋅− ⋅ ⋅

det

= ( )( )( ) ( )( )( ) ( )( )( ) ×++ −

1

212 109 20 4 295 5 574 977 189 208 377 8 312 8 312 370 7 108 567 . [ , ,, , , , . ,

( )( )( ) ( )( ) 8 312 4 295 8 312 108 567 208 377 20 5 574 977 370 7 189 , , , , , ,, . ] − − ( ) ( )( )( )

= 0 406 .

( ) b

nS S S S S S S S S S S S S n S S S

A 2

2 7 162 513 2 25 36 711

= ⋅ ⋅ + ⋅ ⋅ + ⋅ ⋅ − ⋅ ⋅ − ⋅ ⋅− ⋅ ⋅

det

= ( )( )( ) ( )( )( ) ( )( )( ) ×++ −

1

212 109 20 8 312 108 567 370 7 4 295 8 312 189 370 7 208 377 . [ , , ., , . ,

( )( ) 8 312 8 312 189 208 377 4 295 20 108 567 370 7 370 7 , , , , , . .] ( ) − − ( )( )( ) ( )( )( )

= 0 00368 .

The resulting correlation curve is as follows:

í = 0.420 + 0.406x + 0.00368x2

How do I determine the polynomial correlation coefficient?

The polynomial correlation coefficient (r) is calculated using the parameters SP and Sy. The

parameter SP is defined in Equation 11-26 as follows:

S ( ) ny y p ii

i

n

= − − =

∑ 1

3

2

1

$

To calculate SP, the predicted PM concentrations must be determined by substituting the PM

CEMS response values into the correlation curve. For example, the predicted PM concentration

for Run 10 is
Y-9

y$ =+ + = 0 420 0 406 12 0 00368 12 5819 .. . . () () 2

Table Y-3 shows the predicted PM concentrations for all of the 20 test runs used to develop the

correlation. Substitute these values into Equation 11-26 to calculate the value for Sp:

S ( ) ( )( )( ) ( ) p = − − + − + − ++ − =

1

20 3 2 497 21 3717 2 9 3809 51 17 884 185 1951 2 22 2 .... .. . . . K

Table Y-3. Predicted PM Concentrations (í) for Polynomial Correlation

Run no.

PM CEMS 

response, mA

PM concentration,

 mg/acm

Predicted PM

concentration (í)

a

1 4.9 2.1 2.497

2 7.6 2.9 3.717

3 7.8 5.1 3.809

4 9.2 4.8 4.465

5 10.1 2.9 4.894

6 11.06 6.3 5.358

10 12 8.4 5.819

11 18.2 8.4 9.024

12 15.7 4.9 7.698

13 13.9 6.3 6.771

14 19.7 13 9.842

15 21.02 9.1 10.576

16 22.3 13.9 11.299

17 24.1 8.7 12.337

19 24.7 12.6 12.688

20 26.4 15.9 13.698

21 29.08 16.2 15.333

22 29.3 13.5 15.469

23 30.5 15.9 16.221

24 33.1 18.5 17.884

a

 Calculated using the correlation equation: í = 0.420 + 0.406x + 0.00368x2.

The parameter Sy was determined previously to be 5.095. As shown below, substituting Sy and

SP into Equation 11-33 results in a correlation coefficient of 0.924, which satisfies the minimum

value of 0.85 specified by PS-11.

( )

( ) r

S

S

p

y

=− =− = 1 1

1951

5095

0 924

2

2

2

2

.

. .
Y-10

How do I calculate the confidence interval half range percentage for the polynomial correlation?

The half range for the 95 percent CI for the predicted PM concentration must be determined at

the minimum value of the parameter delta ()min). The expression for calculating ) is provided in

Equation 11-25 of PS-11:

∆= + + + + + C Cx C C x Cx Cx 0 1 23 ( ) 2

4

3

5

4 22 2

Before calculating ), it is first necessary to calculate the parameters C0 to C5, which are defined

in Equation 11-23 of PS-11 as follows:

( ) C

SS S

D 0

24 3

2

= ⋅ −

( ) C

SS SS

D 1

32 14 = ⋅−⋅

C

SS S

D 2

13 2

2

= ⋅ −

( ) C

nS S

D 3

4 2

2

= −

( ) C

S S nS

D 4

12 3

= ⋅ −

( ) C

nS S

D 5

2 1

2

= −

The variable D in the denominator of these expressions is determined using Equation 11-24. For

this example, this calculation is as follows:

D nS S S S S S S S S S S S = ⋅− + ⋅−⋅ + ⋅− ( ) ( ) ( ) 24 3

2

13 2 1 4 21 3 2

2

= −+ − + 20 8 312 5 574 977 208 377 370 7 208 377 8 312 370 7 5 574 577 ( ) ( )( ) ( ) (( )( ) ( )( )) 2, ,, , . , , .,,

8 312 370 7 208 377 8 312 2120 10 ( ) ( )( )( )2 9 , ., , . − =×

Substituting the values for D and for the variables S1 to S4, which were defined previously, you

find the following coefficients for determining ):

( ) ( )( ) ( )

C0

3

9

8 312 5 574 977 208 377

212 10

1375 , ,, ,

. . −× =
Y-11

( ) ( )( ) ( )( ) C1 9

208 377 8 312 370 7 5 574 977

212 10 = 01578 −

× = − , , .,,

. .

( ) ( )( )( )

C2

2

9

370 7 208 377 8 312

212 10 = 0 003846 −× = ., ,. .

( ) ( )( )( )

C3

2

9

20 5 574 977 8 312

212 10 = 0 02001 −

× = ,, ,

. .

( ) ( )( ) ( )( ) C4 9

370 7 8 312 20 208 377

212 10 = 0 0005126 −

× = − ., ,

. .

( ) ( )( ) ( )

C5

2

9

5

20 8 312 370 7

212 10 = 1360 10 −

×

= × − , .

. .

When these values are substituted, Equation 11-25 becomes the following:

∆= + − + + + − + × ( ) ( )( ) ( ) ( ) − 1375 2 01578 2 0 003846 0 02001 2 0 0005126 136 10 2 3 54 .. . . . . x x xx

= − + − + × − 1375 0 3155 0 0277 0 001025 136 10 2 3 54 .. . . . x x x x

Table Y-4 summarizes the results when the PM CEMS response values for each test run are

substituted into the above equation.

Table Y-4. Values for )

Run 

no.

PM CEMS 

response, mA

PM concentration,

 mg/acm x2 x3 x4 )

1 4.9 2.1 24.01 118 576 0.3817

2 7.6 2.9 57.76 439 3,336 0.1727

3 7.8 5.1 60.84 475 3,702 0.1634

4 9.2 4.8 84.64 779 7,164 0.1162

5 10.1 2.9 102.01 1,030 10,406 0.0995

6 11.06 6.3 122.32 1,353 14,963 0.0906

10 12.0 8.4 144.00 1,728 20,736 0.0884

11 18.2 8.4 331.24 6,029 109,720 0.1202

12 15.7 4.9 246.49 3,870 60,757 0.1084

13 13.9 6.3 193.21 2,686 37,330 0.0960

14 19.7 13.0 388.09 7,645 150,614 0.1204

15 21.02 9.1 441.84 9,287 195,223 0.1162

16 22.3 13.9 497.29 11,090 247,297 0.1092

(continued)
Table Y-4. (continued)

Run 

no.

PM CEMS 

response, mA

PM concentration,

 mg/acm x2 x3 x4 )

Y-12

17 24.1 8.7 580.81 13,998 337,340 0.0985

19 24.7 12.6 610.09 15,069 372,210 0.0959

20 26.4 15.9 696.96 18,400 485,753 0.0960

21 29.08 16.2 845.65 24,591 715,118 0.1417

22 29.3 13.5 858.49 25,154 737,005 0.1492

23 30.5 15.9 930.25 28,373 865,365 0.2046

24 33.1 18.5 1,095.61 36,265 1,200,361 0.4312

The minimum value for ) is 0.0884 and corresponds to Run 10. Using this value for )min, the

value for SP calculated previously, and the t statistic with n-3 degrees of freedom, you can

calculate the confidence interval half range using Equation 11-27.

CI t S ( )( ) df p =⋅ = = ∆ min 211 1951 0 0884 1223 .. . .

The confidence interval half range as a percentage of the emission limit is calculated using

Equation 11-28 as follows:

CI

CI

EL

% . =× = × = 100% . 1224

20 100% 612%

The resulting value for CI% is 6.12 percent. Because this value is less than10 percent, the

correlation satisfies the criterion for confidence interval half range.

How do I determine the tolerance interval half range percentage for the polynomial correlation?

To calculate the tolerance interval half range, the parameter nN must be calculated using

Equation 11-31 of PS-11.

n′ == = 1 1

0 0884 1131 ∆ min . .

With nN rounded to the nearest integer (11), the values for the variables unN and vdf with n-3

degrees of freedom are taken from Table 1 of PS-11 and substituted into Equation 11-30 to

determine kT:

k uv ( )( ) T n df =⋅= = ′ 1195 1400 1673 .. .

Using this value for kT and the value for SP calculated previously, the tolerance interval half

range is calculated using Equation 11-29 as follows:
Y-13

TI k S ( )( ) =⋅= = T P 1673 1951 3263 .. .

Finally, to determine the tolerance interval as a percentage of the emission limit (TI%),

Equation 11-32 is used as follows:

TI

TI

EL

% .

. =× = × = 100% . 326320 0 100% 16 3%

The resulting value for TI% is 16.3 percent. Because this value is less than 25 percent, the

correlation satisfies the criterion for tolerance interval half range. In summary, the polynomial 

correlation developed from the data satisfies all of the criteria specified in PS-11.

How do I develop a logarithmic correlation curve from the data?

A logarithmic correlation curve is expressed as follows:

y b b x b bx $ =+ =+ ( ) ′ 01 01 ln

Developing a logarithmic correlation entails following the same procedures used to develop a

linear correlation. However, instead of using the PM CEMS response (x) values, the natural

logarithm of the PM CEMS values (xN values) are used in the calculations. Table Y-5

summarizes the test data, including the corresponding values of xN.

Table Y-5. Data for Developing Logarithmic Correlation

Run no.

PM CEMS

response, mA

Logarithm of PM

CEMS response

x' = ln (x) 

PM concentration,

mg/acm

1 4.9 1.589 2.1

2 7.6 2.028 2.9

3 7.8 2.054 5.1

4 9.2 2.219 4.8

5 10.1 2.313 2.9

6 11.06 2.403 6.3

10 12 2.485 8.4

11 18.2 2.901 8.4

12 15.7 2.754 4.9

13 13.9 2.632 6.3

14 19.7 2.981 13

15 21.02 3.045 9.1

(continued)
Table Y-5. (continued)

Run no.

PM CEMS

response, mA

Logarithm of PM

CEMS response

x' = ln (x) 

PM concentration,

mg/acm

Y-14

16 22.3 3.105 13.9

17 24.1 3.182 8.7

19 24.7 3.207 12.6

20 26.4 3.273 15.9

21 29.08 3.370 16.2

22 29.3 3.378 13.5

23 30.5 3.418 15.9

24 33.1 3.500 18.5

Following the same procedures described previously for developing a linear correlation curve,

you can calculate values for the parameters xGN and yG, using Equation 11-5; SxNxN and SxNy, using

Equation 11-7; b1, using Equation 11-6; and b0, using Equation 11-4:

x ′ = + + ++ = ( ) 1

20 1589 2 028 2 054 3500 2 79 ... . . K

y = + + ++ = ( ) 1

20 21 2 9 51 185 9 47 ... . . K

S ( )( )( ) ( ) x x′ ′ = − + − + − ++ − = 1589 2 79 2 028 2 79 2 054 2 79 350 2 79 574 222 2 . . . . . . .. . K

( )( ) ( )( )

( )( )

Sx y′ ′ = − − + − − ++

− −=

1589 2 79 21 9 47 2 028 2 79 2 9 9 47

350 2 79 185 9 47 47 8

. . .. . . . .

. . .. .

K

b1

47 8

574 = = 8 323 .. .

b ( )( ) 0 = − =− 9 47 8 323 2 79 1377 ... .

The resulting equation for the logarithmic correlation curve is

y x $ =− + ⋅ 138 8 32 . . ln( )

How do I determine the logarithmic correlation coefficient?

Determining the logarithmic correlation coefficient requires calculating the predicted PM

concentration (í) for each test run, the parameter SL, and the parameter Sy following the same

procedure used to calculate the linear correlation coefficient. The only difference is that the

predicted PM concentrations are calculated using the logarithmic correlation equation. 

Table Y-6 shows the PM concentrations calculated using the logarithmic correlation coefficient.
Y-15

Table Y-6. Predicted PM Concentrations for the Logarithmic Correlation

Run no.

PM CEMS

response, mA

Logarithm of PM

CEMS response

x' = ln (x) 

PM concentration,

mg/acm

Predicted PM

concentration (í)

a

1 4.9 1.589 2.1 -0.539

2 7.6 2.028 2.9 3.114

3 7.8 2.054 5.1 3.330

4 9.2 2.219 4.8 4.704

5 10.1 2.313 2.9 5.481

6 11.06 2.403 6.3 6.237

10 12 2.485 8.4 6.916

11 18.2 2.901 8.4 10.382

12 15.7 2.754 4.9 9.152

13 13.9 2.632 6.3 8.139

14 19.7 2.981 13 11.041

15 21.02 3.045 9.1 11.581

16 22.3 3.105 13.9 12.073

17 24.1 3.182 8.7 12.719

19 24.7 3.207 12.6 12.924

20 26.4 3.273 15.9 13.478

21 29.08 3.370 16.2 14.283

22 29.3 3.378 13.5 14.345

23 30.5 3.418 15.9 14.679

24 33.1 3.500 18.5 15.360

a

 Calculated using the logarithmic correlation equation: í = !13.8 + 8.32 @ ln(x).

With the values for í shown in Table Y-6, the calculation for SL is as follows:

( ) S ( ) ( )( ) ( ) L = − − − + − ++ − =

1

20 2 0539 21 3114 2 9 1536 185 2 305 22 2 .... .. . K

The value for Sy is the same as was calculated for the linear correlation (5.095). Substituting the

values for SL and Sy into Equation 11-14, the logarithmic correlation coefficient is

( )

( ) r =− = 1

2 305

5095

0892

2

2

.

. .

The logarithmic correlation coefficient (0.892) is greater than the minimum value specified by

PS-11 (0.85) and is acceptable.
Y-16

How do I determine the confidence interval half range percentage for the logarithmic

correlation?

The confidence interval half range for the logarithmic correlation is determined using the same

procedures used for the linear correlation. Substituting the value for SL calculated above (2.305)

and the same value for the t-statistic (2.101) used for the linear correlation, Equation 11-8 yields

CI = = ( )( ) 2101 2 305

1

20 .. . 1083

The emission limit in this example is 20.0 mg/acm. To determine CI as a percentage of the

emission limit, Equation 11-10 is used as follows:

1083

20 100% 541% . × = .

The resulting value for CI% is 5.41 percent. Because this value is less than 10 percent, the

correlation satisfies the criterion for confidence interval half range.

How do I determine the tolerance interval half range percentage for the logarithmic correlation?

The tolerance interval half range for the logarithmic correlation is determined using the same

procedures used for the linear correlation. Substituting the value for SL calculated above (2.305)

and the same value for the kT used for the linear correlation (1.629), Equation 11-11 yields

TI = = (1629 2 305 375 .. . )( )

The emission limit in this example is 20.0 mg/acm. To determine TI as a percentage of the

emission limit, Equation 11-13 is used as follows:

375

20 0 100% 188% .. × = .

The resulting value for TI% is 18.8 percent. Because this value is less than 25 percent, the

correlation satisfies the criterion for confidence interval half range.

How do I develop an exponential correlation curve from the data?

An exponential correlation curve is expressed as follows:

y be $ b x = 0 1

To develop an exponential correlation curve, follow the same procedures used to develop a

linear correlation curve. However, instead of using the PM CEMS concentrations (y) values, use
Y-17

the natural logarithm of the PM CEMS concentrations (yN values) to perform the calculations. 

Table Y-7 summarizes the test data, including the corresponding values of yN.

Table Y-7. Transformed Data for Developing Exponential Correlation

Run no.

PM CEMS

response, mA

PM concentration,

mg/acm

Logarithm of PM

CEMS response

y' = ln (y) 

1 4.9 2.1 0.742

2 7.6 2.9 1.065

3 7.8 5.1 1.629

4 9.2 4.8 1.569

5 10.1 2.9 1.065

6 11.06 6.3 1.841

10 12 8.4 2.128

11 18.2 8.4 2.128

12 15.7 4.9 1.589

13 13.9 6.3 1.841

14 19.7 13 2.565

15 21.02 9.1 2.208

16 22.3 13.9 2.632

17 24.1 8.7 2.163

19 24.7 12.6 2.534

20 26.4 15.9 2.766

21 29.08 16.2 2.785

22 29.3 13.5 2.603

23 30.5 15.9 2.766

24 33.1 18.5 2.918

Following the same procedures described previously for developing a linear correlation curve, 

calculate values for the parameters xG and yGN, using Equation 11-5; Sxx and SxyN

, using

Equation 11-7; b1N, using Equation 11-6; and b0N, using Equation 11-4:

x = + + ++ = ( ) 1

20 4 9 7 6 7 8 331 1853 ... . . K

y ′ = + + ++ = ( ) 1

20 0 742 1065 1629 2 918 2 077 ... . . K

S ( ) ( )( ) ( ) xx = − + − ++ − = 4 9 1853 7 6 1853 331 1853 1 442 2 22 2 . . . . . . ,. K

( )( ) ( )( )

( )( )

Sxy′ = − − + − − ++

− −=

(

) . .. . . .. .

. .. . .

4 9 1853 0 742 2 077 7 6 1853 1065 2 077

331 1853 2 918 2 077 9658

K
Y-18

b1

9658

1442 2 = = 0 0670 .. .

b′ =− = ( )( ) 0 2 077 0 0670 1853 08357 . . ..

The resulting equation is expressed in the following form:

y b bx $′ = ′ + ′ 0 1

For this example, the correlation equation becomes

y x $′ = 0836 0 0670 . . +

Converting this to the format that yields the predicted PM concentration in the correct units

entails taking the exponential of each side of the equation, as follows:

Exp y Exp x ( ) ( )

y e x

$ . .

$ . .

′ = +

=

08357 0 0670

2 31 0 0670

In the resulting correlation equation, the coefficient 2.31 is simply the exponential of b0N.

How do I determine the exponential correlation coefficient?

Determining the exponential correlation coefficient requires calculating the predicted PM

concentration (íN) for each test run, the parameter SL, and the parameter SyN following the same

procedure used to calculate the linear correlation coefficient. Table Y-8 shows the PM

concentrations calculated using the linear form (i.e., íN = b0N + b1Nx) of the exponential

correlation equation.

Table Y-8. Predicted PM Concentrations Using Exponential Correlation Equation

Run no.

PM CEMS response,

mA

PM concentration,

mg/acm

Logarithm of PM

CEMS response

y' = ln (y) 

Predicted PM

concentration (íN)

a

1 4.9 2.1 0.742 1.164

2 7.6 2.9 1.065 1.345

3 7.8 5.1 1.629 1.358

4 9.2 4.8 1.569 1.452

5 10.1 2.9 1.065 1.512

6 11.06 6.3 1.841 1.576

10 12 8.4 2.128 1.639

11 18.2 8.4 2.128 2.055

12 15.7 4.9 1.589 1.887

(continued)
Table Y-8. (continued)

Run no.

PM CEMS response,

mA

PM concentration,

mg/acm

Logarithm of PM

CEMS response

y' = ln (y) 

Predicted PM

concentration (íN)

a

Y-19

13 13.9 6.3 1.841 1.767

14 19.7 13 2.565 2.155

15 21.02 9.1 2.208 2.243

16 22.3 13.9 2.632 2.329

17 24.1 8.7 2.163 2.450

19 24.7 12.6 2.534 2.490

20 26.4 15.9 2.766 2.604

21 29.08 16.2 2.785 2.783

22 29.3 13.5 2.603 2.798

23 30.5 15.9 2.766 2.878

24 33.1 18.5 2.918 3.052

a

 Calculated using the exponential correlation equation: íN = 0.836 + 0.067x. 

Using the values for íN shown in Table Y-8, calculate SL as follows:

( ) S ( ) ( )( ) ( ) L − − − + − ++ − =

1

20 2 1164 0 742 1345 1065 3052 2 918 0 278 22 2 .. .. . . . K

The value for SyN is determined using Equation 11-15 with the values for yN substituted for the

values of y, as follows:

( ) ( )( ) ( )

( ) Sy = − + − ++ −

− = 0 742 2 077 1065 2 077 2 918 2 077

20 1 0 643

22 2 .. .. .. . K

Substitute the values for SL and SyN into Equation 11-14 to calculate the exponential correlation

coefficient:

( )

( ) r =− = 1

0 278

0 643

0 902

2

2

.

. .

The exponential correlation coefficient (0.902) is greater than the minimum value specified by

PS-11 (0.85) and is acceptable.

How do I determine the confidence interval half range percentage for the exponential

correlation?

The confidence interval half range for the exponential correlation is determined using

Equation 11-8, which is the same equation used to calculate the confidence interval half range
1 The equations presented in this section for exponential and power models correspond to the amended

version of PS-11, which was not yet published at the time this was prepared.

Y-20

for linear correlations. Because the confidence interval half range is calculated on the log scale,

it is designated as CIN for the exponential model. Using Equation 11-8, CIN is calculated as

follows:

CI t S ( )( ) 1

n df, 1 a 2 L ′ = ⋅= = − 2101 0 278

1

20 .. . 01307

The lower confidence limit (LCLN) and upper confidence limit (UCLN) are calculated using

Equations 11-40 and 11-41, respectively1

LCLN = yGN - CIN = 2.077 - 0.1307 = 1.946

UCLN = yGN + CIN = 2.077 + 0.1307 = 2.207

Next, the confidence interval half range (CI) must be calculated on the original PM

concentration scale using Equation 11-42 from PS-11

CI

ee ee UCL LCL

= − = − =

′ ′ ( )( ) .

. .

2 2 1045

2 207 1 946

The confidence interval as a percentage of the emission limit is calculated using Equation 11-10

as follows:

CI% . =× = . 1045

20 100% 523%

The resulting value for CI% is 5.23 percent. Because this value is less than 10 percent, the

correlation satisfies the criterion for confidence interval half range.

How do I determine the tolerance interval half range percentage for the exponential correlation?

The tolerance interval half range for the exponential correlation also is calculated using the same

procedure as is used for linear models. Because it is calculated on the log scale, it is denoted

here as TIN. Using Equation 11-11, TIN is calculated as follows:

TIN = kT @ SL = (1.629)(0.278) = 0.453

The lower tolerance limit (LTLN) and upper tolerance limit (UTLN) are calculated using

Equations11-43 and 11-44, respectively

LTLN = yGN - TIN = 2.077 - 0.453 = 1.624

UTLN = yGN + TIN = 2.077 + 0.453 = 2.530
Y-21

Next, the tolerance interval half range (TI) must be calculated on the original PM concentration

scale using Equation 11-45 from PS-11

TI

e e ee UTL LTL

= − = − =

′ ′ ( )( ) .

. .

2 2 374

2 53 1 624

The tolerance interval as a percentage of the emission limit is calculated using Equation 11-13 as

follows:

TI% . =× = . 374

20 100% 18 7%

The resulting value for TI% is 18.7 percent. Because this value is less than 25 percent, the

correlation satisfies the criterion for confidence interval half range. In summary, the exponential

model satisfies the PS-11 criteria for correlation coefficient, confidence interval half range

percentage, and tolerance interval half range percentage.

How do I develop a power correlation curve from the data?

A power correlation curve has the following form:

y bx $ b = 0 1

Developing a power correlation requires transforming both the PM CEMS response (x) values,

as described previously for a logarithmic correlation, and the PM concentration (y) values, as

was described above for an exponential correlation. The procedures used to develop a linear

correlation are then followed using the transformed values for x and y. Table Y-9 shows the test

data and transformed values that must be used.

Table Y-9. Summary of Test Data and Transformed Values for

Developing Power Correlation

Run no.

PM CEMS response,

mA

Logarithm of PM

CEMS response

x' = ln (x) 

PM concentration,

mg/acm

Logarithm of PM

CEMS response

y' = ln (y) 

1 4.9 1.589 2.1 0.742

2 7.6 2.028 2.9 1.065

3 7.8 2.054 5.1 1.629

4 9.2 2.219 4.8 1.569

5 10.1 2.313 2.9 1.065

6 11.06 2.403 6.3 1.841

10 12 2.485 8.4 2.128

11 18.2 2.901 8.4 2.128

(continued)
Table Y-9. (continued)

Run no.

PM CEMS response,

mA

Logarithm of PM

CEMS response

x' = ln (x) 

PM concentration,

mg/acm

Logarithm of PM

CEMS response

y' = ln (y) 

Y-22

12 15.7 2.754 4.9 1.589

13 13.9 2.632 6.3 1.841

14 19.7 2.981 13 2.565

15 21.02 3.045 9.1 2.208

16 22.3 3.105 13.9 2.632

17 24.1 3.182 8.7 2.163

19 24.7 3.207 12.6 2.534

20 26.4 3.273 15.9 2.766

21 29.08 3.370 16.2 2.785

22 29.3 3.378 13.5 2.603

23 30.5 3.418 15.9 2.766

24 33.1 3.500 18.5 2.918

Following the same procedures described previously for developing a linear correlation curve, 

calculate values for the parameters xGN and yGN, using Equation 11-5; SxNxN

 and SxNyN, using

Equation 11-7; b1N, using Equation 11-6; and b0N, using Equation 11-4:

x ′ = + + ++ = ( ) 1

20 1589 2 028 2 054 35 2 792 ... . K

y ′ = + + ++ = ( ) 1

20 0 742 1065 1629 2 918 2 077 ... . . K

S ( ) ( )( ) ( ) x x′ ′ = − + − ++ − = 1589 2 792 2 028 2 792 350 2 792 574 22 2 . . . . .. . K

( )( ) ( )( )

( )( )

Sx y′ ′ = − − + − − ++

− −=

1588 2 792 0 742 2 077 2 028 2 792 1065 2 077

350 2 792 2 918 2 077 6 22

. .. . . .. .

.. . . .

K

b1

6 22

574 = = 10839 .. .

b′ = − =− ( )( ) 0 2 077 10839 2 792 0 9492 ... .

The resulting equation is expressed in the following form:

y b bx $′ = ′ + ′ ′ 0 1

For this example, the correlation equation becomes

y x $′ = − 0 9492 1084 . . + ′

Converting this to the format that yields the predicted PM concentration in the correct units

entails taking the exponential of each side of the equation, as follows:
Y-23

Exp(íN) = Exp(-0.9492 + 1.084xN) 

í = 0.387x1.084

In this equation, the coefficient 0.387 is the exponential of -0.9492, the value for b0N.

How do I determine the power correlation coefficient?

To determine the power correlation coefficient, the predicted PM concentration (íN) must be

calculated for each test run. Table Y-10 shows the predicted PM concentrations calculated using

the linear form (i.e., íN = b0N + b1NxN) of the power correlation equation.

Table Y-10. Predicted PM Concentrations Using the Power Correlation Equation

Run no.

PM CEMS

response, mA

Logarithm of PM

CEMS response

x' = ln (x) 

PM

concentration,

mg/acm

Logarithm of PM

CEMS response

y' = ln (y) 

Predicted PM

concentration (íN)

a

1 4.9 1.589 2.1 0.742 0.773

2 7.6 2.028 2.9 1.065 1.249

3 7.8 2.054 5.1 1.629 1.277

4 9.2 2.219 4.8 1.569 1.456

5 10.1 2.313 2.9 1.065 1.557

6 11.06 2.403 6.3 1.841 1.656

10 12 2.485 8.4 2.128 1.744

11 18.2 2.901 8.4 2.128 2.196

12 15.7 2.754 4.9 1.589 2.035

13 13.9 2.632 6.3 1.841 1.903

14 19.7 2.981 13 2.565 2.281

15 21.02 3.045 9.1 2.208 2.352

16 22.3 3.105 13.9 2.632 2.416

17 24.1 3.182 8.7 2.163 2.500

19 24.7 3.207 12.6 2.534 2.527

20 26.4 3.273 15.9 2.766 2.599

21 29.08 3.370 16.2 2.785 2.704

22 29.3 3.378 13.5 2.603 2.712

23 30.5 3.418 15.9 2.766 2.755

24 33.1 3.500 18.5 2.918 2.844

a

 Calculated using the exponential correlation equation: íN = !0.9492 + 1.084xN
Y-24

CI ′ = = ( )( ) 2101 0 249

1

20 .. . 0117

The parameter SL is calculated using Equation 11-9 with the predicted values (íN) given in the

above table, and the parameter SyN

 is calculated following the same procedure used for the

exponential correlation coefficient, as follows:

( ) S ( ) ( )( ) ( ) L = − − + − ++ − =

1

20 2 0 773 0 742 1249 1065 2 844 2 918 0 249 22 2 .. .. .. . K

( ) ( )( ) ( )

Sy′ = − + − ++ −

− = 0 742 2 077 1065 2 077 2 918 2 077

20 1

0 643

22 2 .. .. .. . K

Substituting the values for SL and SyN into Equation 11-14, the power correlation coefficient is

( )

( ) r =− = 1

0 249

0 643

0 922

2

2

.

. .

The correlation coefficient for the power model (0.922) is greater than the minimum value

specified by PS-11 (0.85) and is acceptable.

How do I determine the confidence interval half range percentage for the power correlation?

The confidence interval half range for the power correlation is determined using the same

procedures used for the exponential correlation. Because the confidence interval half range is

calculated on the log scale, it is designated as CIN for the exponential model. Using

Equation 11-8, CIN is calculated as follows:

The lower confidence limit (LCLN) and upper confidence limit (UCLN) are calculated using

Equations 11-40 and 11-41, respectively

LCLN = yGN - CIN = 2.077 - 0.117 = 1.96

UCLN = yGN + CIN = 2.077 + 0.117 = 2.194

Next, the confidence interval half range (CI) must be calculated on the original PM

concentration scale using Equation 11-42 from PS-11

CI

ee ee UCL LCL

= − = − =

′ ′ ( )( ) .

. .

2 2 0 936

2 194 1 96

The confidence interval as a percentage of the emission limit is calculated using Equation 11-10

as follows:
Y-25

CI% . =×= . 0 936

20 100% 4 68%

The resulting value for CI% is 4.68 percent. Because this value is less than 10 percent, the

correlation satisfies the criterion for confidence interval half range.

How do I determine the tolerance interval half range percentage for the power correlation?

The tolerance interval half range for the power correlation also is calculated using the same

procedure used for the exponential model. Because it is calculated on the log scale, it is denoted

here as TIN. Using Equation 11-11, TIN is calculated as follows:

TIN = kT @ SL = (1.629)(0.249) = 0.406

The lower tolerance limit (LTLN) and upper tolerance limit (UTLN) are calculated using

Equations11-43 and 11-44, respectively

LTLN = yGN - TIN = 2.077 - 0.406 = 1.671

UTLN = yGN + TIN = 2.077 + 0.406 = 2.483

Next, the tolerance interval half range (TI) must be calculated on the original PM concentration

scale using Equation 11-45 from PS-11

TI

ee ee UTL LTL

= − = − =

′ ′ ( )( ) .

. .

2 2 3327

2 483 1 671

The tolerance interval as a percentage of the emission limit is calculated using Equation 11-13 as

follows:

TI% . =×= . 3327

20 100% 16 6%

The resulting value for TI% is 16.6 percent. Because this value is less than 25 percent, the

correlation satisfies the criterion for confidence interval half range. In summary, the power

model satisfies the PS-11 criteria for correlation coefficient, confidence interval half range

percentage, and tolerance interval half range percentage.

Which model is the best fit for the data?

Table Y-11 summarizes the results of the analysis for all of the correlation models. As indicated

in the table, all five models met the acceptance criteria specified in Section 13.2 of PS-11. The

linear model had the highest correlation coefficient and would be considered the best fit for the

data. However, any of the models could be used to show compliance. The complete output for

the spreadsheet for this example is provided in Appendix A.1.
Y-26

Table Y-11. Summary of Results for Facility A Correlation Test

Model type Correlation equation

Correlation

coefficient (r)

Confidence

interval half

range

Tolerance

interval half

range

Acceptance Criteria for PS-11 $ 0.85 # 10% # 25%

Linear í = -0.621 + 0.544x 0.927 4.49% 15.6%

Polynomial í = 0.420 + 0.406x + 0.00368 x2 0.924 6.12% 16.3%

Logarithmic í = -13.8 + 8.32Ln(x) 0.892 5.41% 18.8%

Exponential í = 2.31e0.0670x 0.902 5.23% 18.7%

Power í = 0.387x1.08 0.922 4.68% 16.6%

Y.2 EXAMPLE 2: HOW DO I DEVELOP A CORRELATION WITH PAIRED SAMPLING

TRAIN DATA?

Y.2.1 Correlation Test Scenario

A correlation test is performed on an incinerator at Facility B. The incinerator is equipped with

an in situ PM CEMS that responds to PM concentrations at actual conditions. The emission

limit, when converted to units that are consistent with the PM CEMS sampling conditions, is

24.3 mg/acm. Fifteen test runs are performed using Method 5i with paired sampling trains

(Train A and Train B). No problems were reported for the test. Table Y-12 summarizes the

correlation test data. Figure Y-2 shows a plot of the data.

Table Y-12. Summary of Correlation Test Data for Facility B

Run no.

PM CEMS

response, mA

PM concentration, mg/acm

Sampling Train A Sampling Train B

1 6.4 1.3 1.2

2 7.5 2.3 2.5

3 11.2 2.5 2.8

4 16.4 7.7 7.9

5 29.9 12.9 12.5

6 22.2 7.8 7.3

7 19.5 5.8 5.9

8 27.4 11.7 12.0

9 32.8 20.4 20.9

10 26.5 20.9 21.3

11 33.5 22.3 22.6

12 28.1 18.4 17.58

13 9.6 3.7 3.9

14 5.3 4.2 4.3

15 18.1 6.8 7.1
Y-27

0

5

10

15

20

25

0 5 10 15 20 25 30 35 40

PM CEMS Response, mA

PM Concentration, mg/acm

Figure Y-2. Plot of correlation test data for Facility B.

Should I average the two PM concentrations for each test run?

You should not average the two PM concentrations for any of the test runs. Instead, you should

treat each PM concentration as a separate data point, as specified in Section 12.3 of PS-11. This

means that for each run, the data set includes two PM concentrations and one PM CEMS

response value. In other words, the correlation should be developed as if there were 30 test runs

with one PM concentration for each run. Table Y-13 shows the data in the form that should be

used for developing the correlation.

Before proceeding with the correlation, paired sampling train data should be checked for

precision and bias. Guidance on precision and bias is provided separately. For this example, it

is assumed that the data meet the criteria of precision and bias and are acceptable.
Y-28

Table Y-13. Paired Sampling Train Test Data in Format

for Developing Correlation

Run No.

PM CEMS Response,

mA

PM Concentration,

mg/acm

1A 6.4 1.3

1B 6.4 1.2

2A 7.5 2.3

2B 7.5 2.5

3A 11.2 2.5

3B 11.2 2.8

4A 16.4 7.7

4B 16.4 7.9

5A 29.9 12.9

5B 29.9 12.5

6A 22.2 7.8

6B 22.2 7.3

7A 19.5 5.8

7B 19.5 5.9

8A 27.4 11.7

8B 27.4 12.0

9A 32.8 20.4

9B 32.8 20.9

10A 26.5 20.9

10B 26.5 21.3

11A 33.5 22.3

11B 33.5 22.6

12A 28.1 18.4

12B 28.1 17.8

13A 9.6 3.7

13B 9.6 3.9

14A 5.3 4.2

14B 5.3 4.3

15A 18.1 6.8

15B 18.1 7.1
Y-29

What are the results of the correlation analysis?

Table Y-14 summarizes the results of the analysis. The complete output for the spreadsheet for

this example is provided in Appendix A.2.

Table Y-14. Summary of Results for Facility B Correlation Test

Model type Correlation equation

Correlation

coefficient (r)

Confidence

interval half

range

Tolerance

interval half

range

Acceptance Criteria for PS-11 $ 0.85 # 10% # 25%

Linear í = -3.24 + 0.672x 0.895 4.98% 20.0%

Polynomial í = 2.85 - 0.173x + 0.0220 x2 0.919 5.91% 18.0%

Logarithmic í = -17.7 + 9.81Ln(x) 0.822 6.37% 25.6%

Exponential í = 1.43e0.0828x 0.921 3.80% 15.9%

Power í = 0.201x1.27 0.892 4.41% 18.7%

Which model should I use?

All of the models except the logarithmic model satisfy the acceptance criteria for PS-11. The

logarithmic model fails to meet the minimum allowable correlation coefficient of 0.85. In

addition, the tolerance interval half range percentage for the logarithmic model exceeds the

allowable percentage of 25 percent. Based on the correlation coefficient, the exponential model

is the best fit. However, the linear, polynomial, and power models could also be used.

Y.3 EXAMPLE 3: HOW DO I DEVELOP THE INITIAL CORRELATION?

Y.3.1 Correlation Test Scenario

A correlation test is performed on a hazardous waste incinerator at Facility C. The incinerator is

equipped with an extractive PM CEMS that extracts and heats the samples to 121°C (250°F). 

The incinerator is subject to 40 CFR 63, subpart EEE, and must meet an emission limit of

34.0 mg/dscm corrected to 7 percent oxygen. Fifteen test runs are performed using a Method 5

sampling train. The average stack gas pressure, moisture content, and oxygen content during the

test were 760 millimeters of mercury, 12 percent moisture, and 10 percent oxygen, respectively.

No problems were reported for the test, and all data points will be used for the correlation. 

Table Y-15 summarizes the correlation test data. Figure Y-3 shows a plot of the data.
Y-30

0

5

10

15

20

25

30

35

0.00 5.00 10.00 15.00 20.00 25.00 30.00

PM CEMS Response, mA

PM Concentration, mg/acm

Figure Y-3. Plot of correlation test data for Facility C.

Table Y-15. Summary of Correlation Test Data for Facility C

Run no. PM CEMS response, mA PM concentration, mg/acm

1 5.8 3.5

2 7.6 9.0

3 7.8 7.8

4 16.4 25.2

5 10.1 16.6

6 11.1 14.3

7 12.0 19.6

8 13.2 20.1

9 15.7 23.4

10 17.6 26.7

11 21.7 26.8

12 21.0 28.9

13 22.3 25.7

14 24.1 26.2

15 24.7 26.2
Y-31

Do I need to convert the emission limit to other units for determining compliance with PS-11?

As indicated in Section 13.2 of PS-11, you must convert the emission limit to units that are

consistent with the measurement conditions of your PM CEMS. For this example, you must

convert the emission limit, which is specified in units of mg/dscm at 7 percent oxygen, to match

the conditions under which the PM CEMS is operating. The following calculations show this

conversion:

( ) C C

T

T

P

P SO a

a

std

std

a mc O

@

.

. %

7% 2

1

1

100

13 9

20 9 2

=× × × ⎛

⎝

⎜ ⎞

⎠

⎟

×

− −

where

CS@7%O2 = PM concentration at standard conditions, corrected to 7 percent oxygen

Ta = stack gas temperature at actual conditions, °K

Tstd = stack gas temperature at standard conditions = 293°K

Pstd = stack gas pressure at standard conditions = 760 mm of Hg

Pa = stack gas pressure at actual conditions, mm of Hg

mc = stack gas moisture content, %

%O2 = stack gas oxygen content, %.

( )

( ) 34 0

121 273

293

760

760

1

1

12

100

139

20 9 10 . .. = ×

+

×

⎛

⎝

⎜ ⎞

⎠

⎟ ×

−

⎛

⎝

⎜

⎜

⎜

⎞

⎠

⎟

⎟

⎟

× − Ca

34 0 195 . . = Ca

Ca = 17 45 .

The emission limit in units of concentration at the operating conditions of the PM CEMS is

17.45 mg/acm. This is the value that must be used for determining the confidence and tolerance

interval half range percentages.

What are the results of the correlation analysis?

Table Y-16 summarizes the results of the correlation analysis. Only the logarithmic model

satisfies all of the acceptance criteria for PS-11. The correlation coefficient for the exponential

model (0.812) fails to meet the minimum allowable value of 0.85. The confidence and tolerance

interval half range percentages for the linear model (11.3 and 35.6 percent, respectively) exceed

the limits of 10 and 25 percent, respectively. For the power model, the confidence interval

(14.3 percent) and tolerance interval (46.4 percent) half range percentages also exceed the

maximum allowable values. 
Y-32

Plot of Polynomial Regression Curve

y = -0.101x2 + 4.296x - 18.371

0

5

10

15

20

25

30

35

0.00 5.00 10.00 15.00 20.00 25.00 30.00

PM CEMS Response, mA

PM Concentration, mg/acm

Figure Y-4. Plot of polynomial correlation curve for Facility C.

Table Y-16. Summary of Results for Facility C Correlation Test

Model type Correlation equation

Correlation

coefficient (r)

Confidence

interval half

range

Tolerance

interval half

range

Acceptance Criteria for PS-11 $ 0.85 # 10% # 25%

Linear í = 2.27 + 1.15x 0.898 11.3% 35.6%

Polynomial í = -18.4 + 4.30x - 0.101x2 0.987 5.56% 13.4%

Logarithmic í = -24.4 + 16.8Ln(x) 0.959 7.32% 23.1%

Exponential í = 5.26e0.0785x 0.812 19.9% 66.5%

Power í = 0.747x1.20 0.908 14.3% 46.4%

The polynomial model has the highest correlation coefficient (0.987) and meets the criteria for

confidence and tolerance interval half range percentages. However, Section 12.4(3) of PS-11

specifies that the maximum value for a polynomial correlation curve must occur at a PM CEMS

response value that is at least 1.25 times the highest PM CEMS response value used to develop

the correlation equation. In this example, the maximum PM CEMS response value is 24.7 mA,

and the maximum value for the correlation curve must occur at a PM CEMS response value of at

least 32.4 mA (24.7 × 1.25 = 30.88). However, the maximum value for the polynomial

correlation curve occurs at 21.2 mA, as is evident from Figure Y-4. Therefore, the polynomial

model cannot be used. The complete output for the spreadsheet for this example is provided in

Appendix A.3.
Y-33

Y.4 EXAMPLE 4: WHAT CAN I DO IF MY PM CEMS FAILS THE CORRELATION

TEST?

Y.4.1 Correlation Test Scenario

A correlation test is performed on a boiler at Facility D. The boiler is equipped with an in situ

PM CEMS that responds to PM concentrations at actual conditions. The emission limit, when

converted to units that are consistent with the PM CEMS sampling conditions, is 18.5 mg/acm. 

The correlation test consists of 15 test runs using a single Method 5 sampling train. Table Y-17

summarizes the correlation test data. Figure Y-5 shows a plot of the data.

Table Y-17. Summary of Test Data for Facility D

Run no.

PM CEMS response,

mA

PM concentration,

mg/acm

1 6.7 3.1

2 7.6 7.4

3 7.8 2.1

4 9.2 3.7

5 10.1 6.1

6 11.1 2.8

7 12.0 4.9

8 18.2 8.3

9 13.7 4.5

10 14.6 8.2

11 19.7 13.0

12 18.0 11.1

13 20.3 11.9

14 20.2 10.6

15 19.7 9.2
Y-34

The results of the correlation analysis are summarized in Table Y-18. None of the models satisfy

all of the correlation criteria specified in PS-11. Although all of the models meet the criteria for

confidence and tolerance interval half range percentages, only the polynomial model has an

acceptable correlation coefficient. However, the polynomial correlation curve has a minimum x

value of 7.05 that is greater than the minimum PM CEMS response (6.7 mA for Run 1) value

used to develop the correlation curve. As stated in Section 12.4(3)(iii) of PS-11, a polynomial

correlation curve cannot be used if it has a minimum value that exceeds the minimum PMCEMS response value used to determine the correlation curve. Consequently, the polynomial

correlation also is unacceptable.

Table Y-18. Results of Initial Correlation for Facility D

Model type Correlation equation

Correlation

coefficient (r)

Confidence

interval half

range

Tolerance

interval half

range

Acceptance Criteria for PS-11 $ 0.85 # 10% # 25%

Linear í = -1.20 + 0.598x 0.841 5.80% 18.3%

Polynomial í = 6.19 - 0.615 + 0.0436x2 0.855 7.68% 18.2%

Logarithmic í = -11.8 + 7.37Ln(x) 0.801 6.41% 20.2%

Exponential í = 1.69e0.0934x 0.806 6.41% 21.3%

Power í = 0.308x1.17 0.782 6.76% 22.6%

0

2

4

6

8

10

12

14

0.0 5.0 10.0 15.0 20.0 25.0

PM CEMS Response, mA

PM Concentration, mg/acm

Figure Y-5. Plot of correlation test data for Facility D.
Y-35

Several options are available for meeting the requirements of PS-11, including the following:

# Repeat the correlation test: If you suspect problems with the manual sampling or the

operation of the PM CEMS, you may want to discard all of the test data and conduct

a new correlation test. In such a case, there must be a valid reason for rejecting the

entire data set. For example, you determine that several of the PM samples were

contaminated. You still must report all of the rejected data.

# Conduct additional test runs. You may want to conduct a limited number of

additional test runs (e.g., five additional runs) if it appears that some of the data are

outliers, problems are suspected for some of the runs, or the range of PM

concentrations measured was not adequate to fully characterize the correlation. In

this case, you can develop the correlation based on all of the data (i.e., the initial 15

runs plus the additional runs), or you can reject some of the test runs, provided you

meet the requirements of Section 8.6(3) of PS-11, which allows you to reject up to

five test runs without explanation and additional runs if the basis for rejecting them is

specified in the test method, in PS-11, or in your quality assurance plan.

# Relocate the PM CEMS and conduct a new correlation test: If the sampling location

is not ideal (e.g., PM concentrations are stratified across the stack at the sampling

point), you may want to consider relocating the PM CEMS to another location that is

more suitable. In this case, you must conduct a full correlation test (i.e., at least 15

valid test runs).

# Replace the PM CEMS: Certain types of PM CEMS may not be suitable for specific

stack gas conditions (e.g., an in situ light scattering instrument in a wet stack). If that

is the case, you may decide to install a PM CEMS that uses a different measurement

principle. If so, you must conduct a full correlation test with the new instrument.

# Consider alternative solutions: Section 12.4(4) allows you to petition the regulatory

authority for approval of alternatives to the correlation requirements of PS-11.

For this example, we will assume that you decide to conduct five additional test runs. These

runs are identified as Runs 16 to 20. Table Y-19 shows the results of the additional runs. Figure

Y-6 shows the results of all 20 test runs.

Table Y-19. Results of Additional Test Runs

Run no.

PM CEMS response,

mA

PM concentration,

mg/acm

16 5.2 2.4

17 8.6 4.8

18 14.1 6.2

19 14.5 7.0

20 17.3 8.6
Y-36

Table Y-20 shows the results of the correlation analysis on all 20 test runs. The analysis shows

that the linear and polynomial models meet the correlation criteria of PS-11, and no further

analyses or testing is required. With the additional data included, the minimum value for the

polynomial correlation curve now occurs at 4.2 mA, which is less than the minimum PM CEMS

value used to develop the curve (5.2 mA). However, a review of Figure Y-6 indicates that Run 2

(7.4 mA, 7.4 mg/acm) may be an outlier. You may decide to repeat the correlation analysis with

Run 2 excluded. Table Y-21 shows the results of the analysis without the data from Run 2. In

the revised analysis, all five models meet the criteria of PS-11. The correlation coefficients for

the linear and polynomial models also are considerably greater for the revised analysis. You

have the option of using the results of either analyses. In either case, the polynomial model

provides the best fit; however, any of the models that satisfy all of the correlation criteria in PS11 can be used. 

Table Y-20. Results of Revised Correlation Analysis with All 20 Runs

Model type Correlation equation

Correlation

coefficient (r)

Confidence

interval half

range

Tolerance

interval half

range

Acceptance Criteria for PS-11 $ 0.85 # 10% # 25%

Linear í = -0.905 + 0.573x 0.861 4.25% 14.7%

Polynomial í = 3.94 - 0.263x + 0.0314x2 0.875 5.47% 14.4%

Logarithmic í = -9.79 + 6.57Ln(x) 0.816 4.82% 16.7%

Exponential í = 1.70e0.0935x 0.840 4.52% 16.3%

Power í = 0.367x1.11 0.824 4.73% 17.0%

0

2

4

6

8

10

12

14

0 5 10 15 20 25

PM CEMS Response, mA

PM Concentration, mg/acm

Figure Y-6. Plot of original 15 runs plus 5 additional runs for Facility D.

Run 2
Y-37

Table Y-21. Results of Revised Correlation Analysis with Run 2 Excluded

Model type Correlation equation

Correlation

coefficient (r)

Confidence

interval half

range

Tolerance

interval half

range

Acceptance Criteria for PS-11 $ 0.85 # 10% # 25%

Linear í = -1.88 + 0.629x 0.912 3.63% 12.4%

Polynomial í = 2.19 - 0.067x + 0.026x2 0.921 4.54% 12.1%

Logarithmic í = -11.7 + 7.24Ln(x) 0.867 4.40% 15.0%

Exponential í = 1.41e0.104x 0.909 3.63% 12.6%

Power í = 0.251x1.24 0.894 3.90% 13.6%

The complete output of the spreadsheet for the initial correlation, the correlation with all five

additional test runs, and the correlation with Run 2 excluded are provided in Appendices A.4

through A.6.
Appendix A.1

Complete Results of Example No. 1 –

Initial Correlation Test
CORRELATION TEST PM CEMS AND REFERENCE METHOD TEST DATA

PM CEMS PM concentration Facility: Facility 

A

response mg/acm Location:

Anytown US

A

Run x y Emission Unit: Boiler No. 3

1 4.9 2.1 Test Dates: March 6 to 15, 200

4

2 7.6 2.93 7.8 5.1

Value Units

4 9.2 4.8 Emission limit: 20 mg/acm

5 10.1 2.96 11.1 6.3

10 12.0 8.411 18.2 8.412 15.7 4.913 13.9 6.314 19.7 13.015 21.0 9.116 22.3 13.917 24.1 8.719 24.7 12.620 26.4 15.921 29.1 16.222 29.3 13.523 30.5 15.924 33.1 18.5

02468

101214161820

0.0 5.0 10.0 15.0 20.0 25.0 

PM CEMS Response

PM Concentration

PS-11 Example 1-R 10-27-04.xls Test Data
CALCULATIONS FOR LINEAR CORRELATION

Facility: Facility A Emission Unit: Boiler No. 3

Location: Anytown USA Test Dates: March 6 to 15, 2004

PM CEMS PM concentration

response mg/acm CALCULATED STATISTICAL PARAMETERS

Run x y (xi-x~)2 (yi-y~)2 (xi-x~)(yi-y~) y^ (y^i-yi)2

1 4.9 2.1 185.86 54.32 100.48 2.047 0.0032 7.6 2.9 119.53 43.16 71.83 3.517 0.3813 7.8 5.1 115.20 19.10 46.90 3.626 2.1724 9.2 4.8 87.10 21.81 43.59 4.388 0.1695 10.1 2.9 71.12 43.16 55.40 4.878 3.9146 11.06 6.3 55.85 10.05 23.69 5.401 0.808

10 12 8.4 42.68 1.14 6.99 5.913 6.18511 18.2 8.4 0.11 1.14 0.36 9.289 0.79012 15.7 4.9 8.03 20.88 12.95 7.928 9.16613 13.9 6.3 21.46 10.05 14.69 6.947 0.41914 19.7 13 1.36 12.46 4.12 10.105 8.37915 21.02 9.1 6.19 0.14 -0.92 10.824 2.97316 22.3 13.9 14.19 19.62 16.69 11.521 5.66017 24.1 8.7 30.99 0.59 -4.29 12.501 14.44819 24.7 12.6 38.03 9.80 19.30 12.828 0.05220 26.4 15.9 61.89 41.34 50.58 13.753 4.60821 29.08 16.2 111.24 45.29 70.98 15.213 0.97522 29.3 13.5 115.93 16.24 43.39 15.332 3.35723 30.5 15.9 143.21 41.34 76.95 15.986 0.00724 33.1 18.5 212.20 81.54 131.54 17.401 1.207

PS-11 Example 1-R 10-27-04.xls Linear Calcs
RESULTS OF LINEAR CORRELATION

Facility: Facility A Emission Unit: Boiler No. 3

Location: Anytown USA Test Dates: March 6 to 15, 2004

Correlation equation: y = -0.621 

Variable Equation Value

n Number of data points = 20 Summary of Acceptance Criteria for PS-11

x

~ x~ = 1/n*(Sum of (xi)) = 18.53 Criterion Actual Allowable 

Sxx Sxx = Sum((xi-x~)^2) = 1,442 Correlation coefficient 0.927 > 0.85 

y

~ y~ = 1/n*(Sum of (yi)) = 9.47 Confidence interval 4.49% < 10% 

Syy Syy = Sum((yi-y~)^2) = 493 Tolerance interval 15.6% < 25% Sxy Sxy = Sum((xi-x~)(yi-y~)) = 785.2 * Indicates correlation coefficient is undefined.

b0 b0 = y~-b1x~ = -0.621

b1 b1 = Sxy/Sxx = 0.544

SL SL = SQRT(1/(n-2)(Sum(yi^-yi)^2)) = 1.910

y^mean y^ at mean x value = 9.470

tf tf from table = 2.101

CI CI = tf*SL*SQRT(1/n) = 0.897

EL Emission Limit = 20.0

CI% CI% =CI/EL*100% = 4.49%

n' n' = n = 20

vf vf from table = 1.384

un' un' from table = 1.177

kT kt = un'*vf = 1.629

TI TI = kt*SL = 3.111

TI% TI% = TI/EL*100% = 15.6%

Sy Sy = SQRT(Syy/(n-1)) = 5.095 PS-11 Correlatio

r2 r2 = 1-(SL^2/Sy^2) = 0.859 Calculation Spre

r r = SQRT((1-SL^2/Sy^2)) = 0.927 Version 2-6 

* Indicates correlation coefficient is undefined.

Plot of Linear Regression Line

y = 0.5445x - 0.6207

05

101520

0.0 5.0 10.0 15.0 20.0 25.0 3

PM CEMS Response

PM Concentration

PS-11 Example 1-R 10-27-04.xls Linear Summary 
CALCULATIONS FOR POLYNOMIAL CORRELATION

Facility: Facility A Emission Unit: Boiler No. 3Location: Anytown USA Test Dates: March 6 to 15, 2004

PM CEMS PM concentration

response mg/acm CALCULATED STATISTICAL PARAMETERS

Run x y x2 x3 x4 xy x2y y^ (y^-y)2 delta (y

1 4.9 2.1 24 118 576 10 50 2.497 0.157 0.3817 

2 7.6 2.9 58 439 3,336 22 168 3.717 0.667 0.1727 

3 7.8 5.1 61 475 3,702 40 310 3.809 1.667 0.1634 

4 9.2 4.8 85 779 7,164 44 406 4.465 0.112 0.1162 

5 10.1 2.9 102 1,030 10,406 29 296 4.894 3.975 0.0995 

6 11.06 6.3 122 1,353 14,963 70 771 5.358 0.887 0.0906 

10 12 8.4 144 1,728 20,736 101 1,210 5.819 6.660 0.0884 

11 18.2 8.4 331 6,029 109,720 153 2,782 9.024 0.390 0.1202 

12 15.7 4.9 246 3,870 60,757 77 1,208 7.698 7.828 0.1084 

13 13.9 6.3 193 2,686 37,330 88 1,217 6.771 0.222 0.0960 

14 19.7 13 388 7,645 150,614 256 5,045 9.842 9.972 0.1204 

15 21.02 9.1 442 9,287 195,223 191 4,021 10.576 2.178 0.1162 

16 22.3 13.9 497 11,090 247,297 310 6,912 11.299 6.764 0.1092 

17 24.1 8.7 581 13,998 337,340 210 5,053 12.337 13.229 0.0985 

19 24.7 12.6 610 15,069 372,210 311 7,687 12.688 0.008 0.0959 

20 26.4 15.9 697 18,400 485,753 420 11,082 13.698 4.849 0.0960 

21 29.08 16.2 846 24,591 715,118 471 13,699 15.333 0.752 0.1417 

22 29.3 13.5 858 25,154 737,005 396 11,590 15.469 3.879 0.1492 

23 30.5 15.9 930 28,373 865,365 485 14,791 16.221 0.103 0.2046 

24 33.1 18.5 1,096 36,265 1,200,361 612 20,269 17.884 0.379 0.4312 

PS-11 Example 1-R 10-27-04.xls Polynomial Calcs
RESULTS OF POLYNOMIAL CORRELATION

Facility: Facility A Emission Unit: Boiler No. 3

Location: Anytown USA Test Dates: March 6 to 15, 2004

Variable Equation Value Correlation equation: y = 0.420 

n Number of data points = 20

S1 S1 = Sum (xi) = 371 Summary of Acceptance Criteria for PS-11

S2 S2 = Sum (xi2) = 8,312 Criterion Actual Allowable 

S3 S3 = Sum (xi3) = 208,377 Correlation coefficient 0.924 > 0.85 

S4 S4 = Sum (xi4) = 5,574,977 Confidence interval 6.12% < 10% 

S5 S5 = Sum (yi) = 189 Tolerance interval 16.3% < 25% S6 S6 = Sum (xiyi) = 4,295 * Indicates correlation coefficient is undefined.

S7 S7 = Sum (xi2yi) = 108,567

det A det A = nS2S4-S2S2S2+S1S3S2-S3S3n+S2S1S3-S4S1S1 = 2.120E+09 Check for Correlation Curve Minimum/Maximum

b0 b0 = (S5S2S4+S1S3S7+S2S6S3-S7S2S2-S3S3S5-S4S6S1)/det A = 0.420 Correlation curve minimum point 

b1 b1 = (nS6S4+S5S3S2+S2S1S7-S2S6S2-S7S3n-S4S1S5)/det A = 0.406 Minimum allowable x value 

b2 b2 = (nS2S7+S1S6S2+S5S1S3-S2S2S5-S3S6n-S7S1S1)/det A = 0.00368 Is correlation curve minimum < minimum x value? 

Sp Sp = Sqrt((1/(n-3)Sum of (y^-Y)2) = 1.951 Correlation curve maximum point 

D D = n(S2S4-S32)+S1(S3S2-S1S4)+S2(S1S3-S22) = 2.120E+09 Extrapolation limit for x (125% of maximum x value) 

C0 C0 = (S2S4-S32)/D = 1.375 Is correlation curve maximum > extrapolation limit? 

C1 C1 = (S3S2-S1S4)/D = -0.1578

C2 C2 = (S1S3-S22)/D = 3.846E-03

C3 C3 = (nS4-S22)/D = 2.001E-02C4 C4 = (S1S2-nS3)/D = -5.126E-04

C5 C5 = (nS2-S12)/D = 1.360E-05

tf tf from table = 2.110

EL Emission Limit = 20.0

CI CI = tf*Sp *sqrt(delta-min) = 1.223

CI% CI% = CI/EL*100= 6.12%

vf vf from table = 1.400

un' un' from table = 1.195

n' n' = 1/delta = 11.31

kT kt = un'*vf = 1.673

TI TI = kt*sp = 3.263

TI% TI% = TI/EL *100% = 16.3%

y~ y~ = 1/n*(Sum of (yi)) = 9.47

Sy Sy = sqrt(Sum of ( yi-y~)^2/(n-1) = 5.09

r2 r2 = 1-(Sp^2/Sy^2) = 0.853

r r = sqrt((1-Sp^2/Sy^2)) = 0.924

Max-min? b2 > 0? minimum 

xmax-min y = -b1/2b2 = -55.08 

1.25xmax 41.38 

* Indicates correlation coefficient is undefined.

Plot of Polynomial Regression C

y = 0.0037x2 + 0.4057x + 0.4

02468

101214161820

0.0 5.0 10.0 15.0 20.0 

PM CEMS Respon

PM Concentration

PS-11 Example 1-R 10-27-04.xls Polynomial Summary 
CALCULATIONS FOR LOGARITHMIC CORRELATION

Facility: Facility A Emission Unit: Boiler No. 3

Location: Anytown USA Test Dates: March 6 to 15, 2004

PM CEMS response PM concentration

Measured Transformed mg/acm CALCULATED STATISTICAL PARA

Run x x' = ln (x) y (xi'-x'~)2 (yi-y~)2 (x'i-x'~)(yi-y~) y^ 

1 4.9 1.589 2.1 1.446 54.317 8.863 -0.2 7.6 2.028 2.9 0.583 43.165 5.017 33 7.8 2.054 5.1 0.544 19.097 3.224 34 9.2 2.219 4.8 0.328 21.809 2.674 45 10.1 2.313 2.9 0.230 43.165 3.149 56 11.06 2.403 6.3 0.151 10.049 1.231 6

10 12 2.485 8.4 0.094 1.145 0.328 611 18.2 2.901 8.4 0.012 1.145 -0.117 1012 15.7 2.754 4.9 0.001 20.885 0.174 913 13.9 2.632 6.3 0.026 10.049 0.507 814 19.7 2.981 13 0.036 12.461 0.666 1115 21.02 3.045 9.1 0.064 0.137 -0.094 1116 22.3 3.105 13.9 0.098 19.625 1.386 1217 24.1 3.182 8.7 0.152 0.593 -0.301 1219 24.7 3.207 12.6 0.172 9.797 1.299 1220 26.4 3.273 15.9 0.232 41.345 3.096 1321 29.08 3.370 16.2 0.334 45.293 3.891 1422 29.3 3.378 13.5 0.343 16.241 2.361 1423 30.5 3.418 15.9 0.392 41.345 4.025 1424 33.1 3.500 18.5 0.501 81.541 6.391 15

PS-11 Example 1-R 10-27-04.xls Logarithmic Calcs
RESULTS OF LOGARITHMIC CORRELATION

Facility: Facility A Emission Unit: Boiler No. 

Location: Anytown USA Test Dates: March 6 to 

Correlation equation: y = -13.766 + 

Variable Equation Value

n Number of data points = 20 Summary of Acceptance Criteria for PS-11

x'~ x'~ = 1/n*(Sum of (x'i)) = 2.79 Criterion Actual Allow

Sx'x' Sx'x' = Sum((x'i-x'~)^2) = 5.74 Correlation coefficient 0.892 > 0

y

~ y~ = 1/n*(Sum of (yi)) = 9.5 Confidence interval 5.41% < 1

Syy Syy = Sum((yi-y~)^2) = 493 Tolerance interval 18.8% < 25

Sx'y Sx'y = Sum((x'i-x'~)(yi-y~)) = 47.8 * Indicates correlation coefficient is undefined.

b0 b0 = y~-b1x'~ = -13.766

b1 b1 = Sx'y/Sx'x' = 8.323

SL SL = SQRT(1/(n-2)(Sum(yi^-yi)^2)) = 2.305

y^mean y^ at mean x value = 9.470

tf tf from table = 2.101

CI CI = tf*SL*SQRT(1/n) = 1.083

EL Emission Limit = 20.0

CI% CI% =CI/EL*100% = 5.41%

n' n' = n = 20

vf vf from table = 1.384

un' un' from table = 1.177

kT kt = un'*vf = 1.629

TI TI = kt*SL = 3.75

TI% TI% = TI/EL*100% = 18.8%

Sy Sy = SQRT(Syy/(n-1)) = 5.095 

r2 r2 = 1-(SL^2/Sy^2) = 0.795 

r r = SQRT((1-SL^2/Sy^2)) = 0.892 

* Indicates correlation coefficient is undefined.

Plot of Logarithmic Regression Cu

y = 8.3229Ln(x) - 13.766

-5

05

101520

0.0 5.0 10.0 15.0 20.0 

PM CEMS Response

PM Concentration

PS-11 Example 1-R 10-27-04.xls Logarithmic Summary 
CALCULATIONS FOR EXPONENTIAL CORRELATION

Facility: Facility A Emission Unit: Boiler No. 

Location: Anytown USA Test Dates: March 6 to 

PM concentration

PM CEMS mg/acm

response Measured Transformed CALCULATED STATISTICAL PARAMETE

Run x y y' = Ln(y) (xi-x~)2 (y'i-y'~)2 (xi-x~)(y'i-y'~) y'^ 

1 4.9 2.1 0.742 185.86 1.782 18.198 1.164 2 7.6 2.9 1.065 119.53 1.024 11.065 1.345 3 7.8 5.1 1.629 115.20 0.200 4.804 1.358 4 9.2 4.8 1.569 87.10 0.258 4.743 1.452 5 10.1 2.9 1.065 71.12 1.024 8.535 1.512 6 11.06 6.3 1.841 55.85 0.056 1.766 1.576 

10 12 8.4 2.128 42.68 0.003 -0.336 1.639 11 18.2 8.4 2.128 0.11 0.003 -0.017 2.055 12 15.7 4.9 1.589 8.03 0.238 1.381 1.887 13 13.9 6.3 1.841 21.46 0.056 1.095 1.767 14 19.7 13 2.565 1.36 0.238 0.570 2.155 15 21.02 9.1 2.208 6.19 0.017 0.327 2.243 16 22.3 13.9 2.632 14.19 0.308 2.091 2.329 17 24.1 8.7 2.163 30.99 0.007 0.482 2.450 19 24.7 12.6 2.534 38.03 0.209 2.818 2.490 20 26.4 15.9 2.766 61.89 0.475 5.424 2.604 21 29.08 16.2 2.785 111.24 0.502 7.469 2.783 22 29.3 13.5 2.603 115.93 0.277 5.662 2.798 23 30.5 15.9 2.766 143.21 0.475 8.251 2.878 24 33.1 18.5 2.918 212.20 0.707 12.250 3.052 

PS-11 Example 1-R 10-27-04.xls Exponential Calcs
RESULTS OF EXPONENTIAL CORRELATION

Facility: Facility A Emission Unit: Boiler No. 3

Location: Anytown USA Test Dates: March 6 to 15, 2004

Correlation equation: y = 2.306 e 0.0

Variable Equation Value

n Number of data points = 20 Summary of Acceptance Criteria for PS-11

x

~ x~ = 1/n*(Sum of (xi)) = 18.53 Criterion Actual Allowable 

Sxx Sxx = Sum((xi-x~)^2) = 1,442.2 Correlation coefficient 0.902 > 0.85 

y'~ y'~ = 1/n*(Sum of (y'i)) = 2.077 Confidence interval 5.23% < 10% 

Sy'y' Sy'y' = Sum((y'i-y'~)^2) = 7.86 Tolerance interval 18.7% < 25% 

Sxy' Sxy' = Sum((xi-x~)(y'i-y'~)) = 96.58 * Indicates correlation coefficient is undefined.

b0' b0' = y'~-b1x~ = 0.836

b0 b0 = e^b0' = 2.306

b1 b1 = Sxy'/Sxx = 0.067

SL SL = SQRT(1/(n-2)(Sum(y'i^-y'i)^2)) = 0.278

y'^mean y'^ at mean x value = 2.077

tf tf from table = 2.101

CI' CI' = tf*SL*SQRT(1/n) = 0.1307

LCL' LCL' = y~' - CI' 1.946

UCL' UCL' = y~' + CI' 2.207

CI CI = (e^UCL' - e^LCL')/2 1.045

EL Emission Limit = 20.0

CI% CI% =CI/EL*100% = 5.23%

n' n' = n = 20

vf vf from table = 1.384

un' un' from table = 1.177

kT kT = un'*vf = 1.629

TI' TI' = kt*SL = 0.453

LTL' LTL' = y~' - TI' 1.624

UTL' UTL' = y~' + TI' 2.530

TI TI = (e^UTL' - e^LTL')/2 3.740

TI% TI% = TI/EL*100% = 18.7%

Sy' Sy' = SQRT(Sy'y'/(n-1)) = 0.643 PS-11 Correlation 

r2 r2 = 1-(SL^2/Sy'^2) = 0.813 Calculation Spread

r r = SQRT((1-SL^2/Sy'^2)) = 0.902 Version 2-6 10/2

* Indicates correlation coefficient is undefined.

Plot of Exponential Regression Curve

y = 2.3064e0.067x

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PM CEMS Response

PM Concentration

PS-11 Example 1-R 10-27-04.xls Exponential Summary 
CALCULATIONS FOR POWER CORRELATION

Facility: Facility A Emission Unit: Boiler No. 3

Location: Anytown USA Test Dates: March 6 to 15, 20

PM concentration

PM CEMS response mg/acm

Measured Transformed Measured Transformed CALCULATED STATISTICAL PARA

Run x x' = Ln(x) y y' = Ln(y) (x'i-x'~)2 (y'i-y'~)2 (x'i-x'~)(y'i-y'~) 

1 4.9 1.589 2.1 0.742 1.446 1.782 1.605 2 7.6 2.028 2.9 1.065 0.583 1.024 0.773 3 7.8 2.054 5.1 1.629 0.544 0.200 0.330 4 9.2 2.219 4.8 1.569 0.328 0.258 0.291 5 10.1 2.313 2.9 1.065 0.230 1.024 0.485 6 11.06 2.403 6.3 1.841 0.151 0.056 0.092 

10 12 2.485 8.4 2.128 0.094 0.003 -0.016 11 18.2 2.901 8.4 2.128 0.012 0.003 0.006 12 15.7 2.754 4.9 1.589 0.001 0.238 0.019 13 13.9 2.632 6.3 1.841 0.026 0.056 0.038 14 19.7 2.981 13 2.565 0.036 0.238 0.092 15 21.02 3.045 9.1 2.208 0.064 0.017 0.033 16 22.3 3.105 13.9 2.632 0.098 0.308 0.174 17 24.1 3.182 8.7 2.163 0.152 0.007 0.034 19 24.7 3.207 12.6 2.534 0.172 0.209 0.190 20 26.4 3.273 15.9 2.766 0.232 0.475 0.332 21 29.08 3.370 16.2 2.785 0.334 0.502 0.410 22 29.3 3.378 13.5 2.603 0.343 0.277 0.308 23 30.5 3.418 15.9 2.766 0.392 0.475 0.432 24 33.1 3.500 18.5 2.918 0.501 0.707 0.595 

PS-11 Example 1-R 10-27-04.xls Power Calcs
RESULTS OF POWER CORRELATION

Facility: Facility A Emission Unit: Boiler No. 3

Location: Anytown USA Test Dates: March 6 to 15, 2004

Correlation equation: y = 0.3870 x 1.0839

Variable Equation Value

n Number of data points = 20 Summary of Acceptance Criteria for PS-11

x'~ x'~ = 1/n*(Sum of (x'i)) = 2.792 Criterion Actual Allowable 

Sx'x' Sx'x' = Sum((x'i-x'~)^2) = 5.74 Correlation coefficient 0.922 > 0.85 

y'~ y'~ = 1/n*(Sum of (y'i)) = 2.077 Confidence interval 4.68% < 10% 

Sy'y' Sy'y' = Sum((y'i-y'~)^2) = 7.86 Tolerance interval 16.6% < 25% Sx'y' Sx'y' = Sum((x'i-x'~)(y'i-y'~)) = 6.22 * Indicates correlation coefficient is undefined.

b0' b0' = y'~-b1x~ = -0.949

b0 b0 = e^b0' = 0.387

b1 b1 = Sx'y'/Sx'x' = 1.084

SL SL = SQRT(1/(n-2)(Sum(y'i^-y'i)^2)) = 0.249

y'^mean y'^ at mean x value = 2.077

tf tf from table = 2.101

CI' CI = tf*SL*SQRT(1/n) = 0.117

LCL' LCL' = y~' - CI' 1.960

UCL' UCL' = y~' + CI' 2.194

CI CI = (e^UCL' - e^LCL')/2 0.936

EL Emission Limit = 20.0

CI% CI% =CI/EL*100% = 4.68%

n' n' = n = 20

vf vf from table = 1.384

un' un' from table = 1.177

kT kT = un'*vf = 1.629

TI' TI' = kTSL = 0.406

LTL' LTL' = y~' - TI' 1.671

UTL' UTL' = y~' + TI' 2.483

TI TI = (e^UTL' - e^LTL')/2 3.327

TI% TI% = TI/EL*100% = 16.6%

Sy' Sy' = SQRT(Sy'y'/(n-1)) = 0.643 PS-11 

r2 r2 = 1-(SL^2/Sy'^2) = 0.850 Calcula

r r = SQRT((1-SL^2/Sy'^2)) = 0.922 Version 

* Indicates correlation coefficient is undefined.

Plot of Power Function Regression Curve

y = 0.387x1.0839

02468

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PM CEMS Response

PM Concentration

PS-11 Example 1-R 10-27-04.xls Power Summary 
SELECTION OF BEST MODEL

Facility: Facility A Emission Unit: Boiler No. 3

Location: Anytown USA Test Dates: March 6 to 15, 

Model Correlation coefficient (a) > 0.85 ? Confidence interval half range percentage < 10% ? Tolerance interval half range percentage < 25% ? Min/max with

allowable rang

Linear 0.927 Yes 4.49% Yes 15.6% Yes (b) 

Polynomial 0.924 Yes 6.12% Yes 16.3% Yes Yes 

Logarithmic 0.892 Yes 5.41% Yes 18.8% Yes (b) 

Exponential 0.902 Yes 5.23% Yes 18.7% Yes (b) 

Power 0.922 Yes 4.68% Yes 16.6% Yes (b) 

(a) * indicates correlation coefficient is undefined; model does not satisfy criterion.(b) Not applicable; criterion applies only to polynomial model.

Best mod

PS-11 Correlation TestCalculation SpreadsheetVersion 2-6 10/25/04

PS-11 Example 1-R 10-27-04.xls Best Model 
Appendix A.2

Complete Results of Example No. 2 –

Initial Correlation Test
CORRELATION TEST PM CEMS AND REFERENCE METHOD TEST DATA

PM CEMS PM concentration Facility: Facility B

response mg/acm Location:

Anytown US

A

Run x y Emission Unit: Incinerator

1A 6.4 1.3 Test Dates:

April 3 to 6, 200

3

1B 6.4 1.22A 7.5 2.3

Value Units

2B 7.5 2.5 Emission limit: 24.3 mg/acm3A 11.2 2.53B 11.2 2.84A 16.4 7.74B 16.4 7.95A 29.9 12.95B 29.9 12.56A 22.2 7.86B 22.2 7.37A 19.5 5.87B 19.5 5.98A 27.4 11.78B 27.4 129A 32.8 20.49B 32.8 20.9

10A 26.5 20.910B 26.5 21.311A 33.5 22.311B 33.5 22.612A 28.1 18.412B 28.1 17.813A 9.6 3.713B 9.6 3.914A 5.3 4.214B 5.3 4.315A 18.1 6.815B 18.1 7.1

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PM CEMS Response

PM Concentration

PS-11 Example 2-R 10-27-04.xls Test Data
CALCULATIONS FOR LINEAR CORRELATION

Facility: Facility B Emission Unit: IncineratorLocation: Anytown USA Test Dates: April 3 to 6, 200

PM CEMS PM concentration

response mg/acm CALCULATED STATISTICAL PARAMETER

Run x y (xi-x~)2 (yi-y~)2 (xi-x~)(yi-y~) y^ 

1A 6.4 1.3 174.94 74.94 114.50 1.065 1B 6.4 1.2 174.94 76.68 115.82 1.065 2A 7.5 2.3 147.06 58.62 92.85 1.805 2B 7.5 2.5 147.06 55.60 90.42 1.805 3A 11.2 2.5 71.01 55.60 62.83 4.292 3B 11.2 2.8 71.01 51.22 60.31 4.292 4A 16.4 7.7 10.41 5.09 7.28 7.788 4B 16.4 7.9 10.41 4.23 6.64 7.788 5A 29.9 12.9 105.54 8.66 30.24 16.863 5B 29.9 12.5 105.54 6.47 26.13 16.863 6A 22.2 7.8 6.62 4.65 -5.55 11.687 6B 22.2 7.3 6.62 7.06 -6.84 11.687 7A 19.5 5.8 0.02 17.28 0.53 9.872 7B 19.5 5.9 0.02 16.46 0.51 9.872 8A 27.4 11.7 60.42 3.04 13.55 15.182 8B 27.4 12 60.42 4.18 15.88 15.182 9A 32.8 20.4 173.54 109.06 137.57 18.812 9B 32.8 20.9 173.54 119.76 144.16 18.812 

10A 26.5 20.9 47.24 119.76 75.22 14.577 10B 26.5 21.3 47.24 128.67 77.97 14.577 11A 33.5 22.3 192.47 152.36 171.24 19.283 11B 33.5 22.6 192.47 159.85 175.41 19.283 12A 28.1 18.4 71.80 71.29 71.54 15.653 12B 28.1 17.8 71.80 61.52 66.46 15.653 13A 9.6 3.7 100.53 39.15 62.73 3.216 13B 9.6 3.9 100.53 36.68 60.73 3.216 14A 5.3 4.2 205.25 33.14 82.47 0.326 14B 5.3 4.3 205.25 32.00 81.04 0.326 15A 18.1 6.8 2.33 9.96 4.82 8.930 15B 18.1 7.1 2.33 8.16 4.36 8.930 

PS-11 Example 2-R 10-27-04.xls Linear Calcs
RESULTS OF LINEAR CORRELATION

Facility: Facility B Emission Unit: IncineratorLocation: Anytown USA Test Dates: April 3 to 6, 2003

Correlation equation: y = -3.237 

Variable Equation Value

n Number of data points = 30 Summary of Acceptance Criteria for PS-11

x

~ x~ = 1/n*(Sum of (xi)) = 19.63 Criterion Actual Allowable 

Sxx Sxx = Sum((xi-x~)^2) = 2,738 Correlation coefficient 0.895 > 0.85 

y

~ y~ = 1/n*(Sum of (yi)) = 9.96 Confidence interval 4.98% < 10% 

Syy Syy = Sum((yi-y~)^2) = 1,531 Tolerance interval 20.0% < 25% Sxy Sxy = Sum((xi-x~)(yi-y~)) = 1,840.8 * Indicates correlation coefficient is undefined.

b0 b0 = y~-b1x~ = -3.237

b1 b1 = Sxy/Sxx = 0.672

SL SL = SQRT(1/(n-2)(Sum(yi^-yi)^2)) = 3.238

y^mean y^ at mean x value = 9.957

tf tf from table = 2.048

CI CI = tf*SL*SQRT(1/n) = 1.211

EL Emission Limit = 24.3

CI% CI% =CI/EL*100% = 4.98%

n' n' = n = 30

vf vf from table = 1.286

un' un' from table = 1.168

kT kt = un'*vf = 1.503

TI TI = kt*SL = 4.866

TI% TI% = TI/EL*100% = 20.0%

Sy Sy = SQRT(Syy/(n-1)) = 7.266 PS-11 Correla

r2 r2 = 1-(SL^2/Sy^2) = 0.801 Calculation Sp

r r = SQRT((1-SL^2/Sy^2)) = 0.895 Version 2-6 

* Indicates correlation coefficient is undefined.

Plot of Linear Regression Line

y = 0.6722x - 3.2371

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PM CEMS Response

PM Concentration

PS-11 Example 2-R 10-27-04.xls Linear Summary 
CALCULATIONS FOR POLYNOMIAL CORRELATION

Facility: Facility B Emission Unit: IncineratorLocation: Anytown USA Test Dates: April 3 to 6, 2003

PM CEMS PM concentration

response mg/acm CALCULATED STATISTICAL PARAMETERS

Run x y x2 x3 x4 xy x2y y^ (y^-y)2 delta 

1A 6.4 1.3 41 262 1,678 8 53 2.649 1.819 0.1326 1B 6.4 1.2 41 262 1,678 8 49 2.649 2.099 0.1326 2A 7.5 2.3 56 422 3,164 17 129 2.795 0.245 0.1009 2B 7.5 2.5 56 422 3,164 19 141 2.795 0.087 0.1009 3A 11.2 2.5 125 1,405 15,735 28 314 3.680 1.392 0.0646 3B 11.2 2.8 125 1,405 15,735 31 351 3.680 0.774 0.0646 4A 16.4 7.7 269 4,411 72,339 126 2,071 5.942 3.090 0.0852 4B 16.4 7.9 269 4,411 72,339 130 2,125 5.942 3.833 0.0852 5A 29.9 12.9 894 26,731 799,254 386 11,533 17.379 20.061 0.0756 5B 29.9 12.5 894 26,731 799,254 374 11,175 17.379 23.804 0.0756 

6A 22.2 7.8 493 10,941 242,891 173 3,844 9.872 4.292 0.0823 6B 22.2 7.3 493 10,941 242,891 162 3,598 9.872 6.614 0.0823 7A 19.5 5.8 380 7,415 144,590 113 2,205 7.858 4.236 0.0906 7B 19.5 5.9 380 7,415 144,590 115 2,243 7.858 3.834 0.0906 8A 27.4 11.7 751 20,571 563,641 321 8,784 14.655 8.732 0.0593 8B 27.4 12 751 20,571 563,641 329 9,009 14.655 7.049 0.0593 9A 32.8 20.4 1,076 35,288 1,157,432 669 21,947 20.884 0.234 0.1573 9B 32.8 20.9 1,076 35,288 1,157,432 686 22,485 20.884 0.000 0.1573 

10A 26.5 20.9 702 18,610 493,155 554 14,677 13.742 51.239 0.0604 

10B 26.5 21.3 702 18,610 493,155 564 14,958 13.742 57.125 0.0604 

11A 33.5 22.3 1,122 37,595 1,259,445 747 25,026 21.785 0.265 0.1920 

11B 33.5 22.6 1,122 37,595 1,259,445 757 25,363 21.785 0.664 0.1920 12A 28.1 18.4 790 22,188 623,484 517 14,529 15.390 9.060 0.0605 12B 28.1 17.8 790 22,188 623,484 500 14,055 15.390 5.808 0.0605 13A 9.6 3.7 92 885 8,493 36 341 3.223 0.227 0.0700 13B 9.6 3.9 92 885 8,493 37 359 3.223 0.458 0.0700 14A 5.3 4.2 28 149 789 22 118 2.556 2.704 0.1785 14B 5.3 4.3 28 149 789 23 121 2.556 3.043 0.1785 15A 18.1 6.8 328 5,930 107,328 123 2,228 6.940 0.020 0.0901 15B 18.1 7.1 328 5,930 107,328 129 2,326 6.940 0.025 0.0901 

PS-11 Example 2-R 10-27-04.xls Polynomial Calcs
RESULTS OF POLYNOMIAL CORRELATION

Facility: Facility B Emission Unit: IncineratorLocation: Anytown USA Test Dates: April 3 to 6, 2003

Variable Equation Value Correlation equation: y = 2.854 

n Number of data points = 30

S1 S1 = Sum (xi) = 589 Summary of Acceptance Criteria for PS-11

S2 S2 = Sum (xi2) = 14,295 Criterion Actual Allowable 

S3 S3 = Sum (xi3) = 385,603 Correlation coefficient 0.919 > 0.85 

S4 S4 = Sum (xi4) = 10,986,838 Confidence interval 5.91% < 10% 

S5 S5 = Sum (yi) = 299 Tolerance interval 18.0% < 25% S6 S6 = Sum (xiyi) = 7,703 * Indicates correlation coefficient is undefined.

S7 S7 = Sum (xi2yi) = 216,158

det A det A = nS2S4-S2S2S2+S1S3S2-S3S3n+S2S1S3-S4S1S1 = 1.198E+10 Check for Correlation Curve Minimum/Maximum

b0 b0 = (S5S2S4+S1S3S7+S2S6S3-S7S2S2-S3S3S5-S4S6S1)/det A = 2.854 Correlation curve minimum point 

b1 b1 = (nS6S4+S5S3S2+S2S1S7-S2S6S2-S7S3n-S4S1S5)/det A = -0.173 Minimum allowable x value 

b2 b2 = (nS2S7+S1S6S2+S5S1S3-S2S2S5-S3S6n-S7S1S1)/det A = 0.02203 Is correlation curve minimum < minimum x value? 

Sp Sp = Sqrt((1/(n-3)Sum of (y^-Y)2) = 2.873 Correlation curve maximum point 

D D = n(S2S4-S32)+S1(S3S2-S1S4)+S2(S1S3-S22) = 1.198E+10 Extrapolation limit for x (125% of maximum x value) 

C0 C0 = (S2S4-S32)/D = 0.698 Is correlation curve maximum > extrapolation limit? 

C1 C1 = (S3S2-S1S4)/D = -0.0799

C2 C2 = (S1S3-S22)/D = 1.895E-03

C3 C3 = (nS4-S22)/D = 1.045E-02C4 C4 = (S1S2-nS3)/D = -2.630E-04

C5 C5 = (nS2-S12)/D = 6.855E-06

tf tf from table = 2.052

EL Emission Limit = 24.3

CI CI = tf*Sp *sqrt(delta-min) = 1.436

CI% CI% = CI/EL*100= 5.91%

vf vf from table = 1.293

un' un' from table = 1.181

n' n' = 1/delta = 16.86

kT kt = un'*vf = 1.527

TI TI = kt*sp = 4.385

TI% TI% = TI/EL *100% = 18.0%

y~ y~ = 1/n*(Sum of (yi)) = 9.96

Sy Sy = sqrt(Sum of ( yi-y~)^2/(n-1) = 7.27

r2 r2 = 1-(Sp^2/Sy^2) = 0.844

r r = sqrt((1-Sp^2/Sy^2)) = 0.919

Max-min? b2 > 0? minimum 

xmax-min y = -b1/2b2 = 3.93 

1.25xmax 41.88 

* Indicates correlation coefficient is undefined.

Plot of Polynomial Regression C

y = 0.022x2 - 0.1731x + 2.8539

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PM CEMS Respon

PM Concentration

PS-11 Example 2-R 10-27-04.xls Polynomial Summary 
CALCULATIONS FOR LOGARITHMIC CORRELATION

Facility: Facility B Emission Unit: IncineratorLocation: Anytown USA Test Dates: April 3 to 6, 2003

PM CEMS response PM concentration

Measured Transformed mg/acm CALCULATED STATISTICAL PARAMET

Run x x' = ln (x) y (xi'-x'~)2 (yi-y~)2 (x'i-x'~)(yi-y~) y^ 

1A 6.4 1.856 1.3 0.929 74.938 8.345 0.504 1B 6.4 1.856 1.2 0.929 76.679 8.441 0.504 2A 7.5 2.015 2.3 0.649 58.625 6.166 2.059 2B 7.5 2.015 2.5 0.649 55.602 6.005 2.059 3A 11.2 2.416 2.5 0.163 55.602 3.015 5.992 3B 11.2 2.416 2.8 0.163 51.218 2.894 5.992 4A 16.4 2.797 7.7 0.001 5.093 0.052 9.732 4B 16.4 2.797 7.9 0.001 4.230 0.047 9.732 5A 29.9 3.398 12.9 0.334 8.663 1.700 15.621 5B 29.9 3.398 12.5 0.334 6.469 1.469 15.621 6A 22.2 3.100 7.8 0.078 4.651 -0.604 12.701 6B 22.2 3.100 7.3 0.078 7.058 -0.743 12.701 7A 19.5 2.970 5.8 0.023 17.278 -0.624 11.429 7B 19.5 2.970 5.9 0.023 16.457 -0.609 11.429 8A 27.4 3.311 11.7 0.240 3.039 0.855 14.765 8B 27.4 3.311 12 0.240 4.175 1.002 14.765 9A 32.8 3.490 20.4 0.449 109.063 6.999 16.529 9B 32.8 3.490 20.9 0.449 119.757 7.334 16.529 

10A 26.5 3.277 20.9 0.209 119.757 5.000 14.437 10B 26.5 3.277 21.3 0.209 128.671 5.183 14.437 11A 33.5 3.512 22.3 0.478 152.358 8.533 16.736 11B 33.5 3.512 22.6 0.478 159.854 8.740 16.736 12A 28.1 3.336 18.4 0.266 71.290 4.353 15.012 12B 28.1 3.336 17.8 0.266 61.518 4.043 15.012 13A 9.6 2.262 3.7 0.312 39.146 3.494 4.480 13B 9.6 2.262 3.9 0.312 36.683 3.382 4.480 14A 5.3 1.668 4.2 1.328 33.139 6.635 -1.346 14B 5.3 1.668 4.3 1.328 31.998 6.519 -1.346 15A 18.1 2.896 6.8 0.006 9.965 -0.239 10.699 15B 18.1 2.896 7.1 0.006 8.161 -0.216 10.699 

PS-11 Example 2-R 10-27-04.xls Logarithmic Calcs
RESULTS OF LOGARITHMIC CORRELATION

Facility: Facility B Emission Unit: IncineratoLocation: Anytown USA Test Dates: April 3 to 

Correlation equation: y = -17.700 

Variable Equation Value

n Number of data points = 30 Summary of Acceptance Criteria for PS-11

x'~ x'~ = 1/n*(Sum of (x'i)) = 2.82 Criterion Actual Allo

Sx'x' Sx'x' = Sum((x'i-x'~)^2) = 10.93 Correlation coefficient 0.822 > 0

y

~ y~ = 1/n*(Sum of (yi)) = 10.0 Confidence interval 6.37% < 

Syy Syy = Sum((yi-y~)^2) = 1,531 Tolerance interval 25.6% < 2

Sx'y Sx'y = Sum((x'i-x'~)(yi-y~)) = 107.2 * Indicates correlation coefficient is undefined.

b0 b0 = y~-b1x'~ = -17.700

b1 b1 = Sx'y/Sx'x' = 9.807

SL SL = SQRT(1/(n-2)(Sum(yi^-yi)^2)) = 4.141

y^mean y^ at mean x value = 9.957

tf tf from table = 2.048

CI CI = tf*SL*SQRT(1/n) = 1.549

EL Emission Limit = 24.3

CI% CI% =CI/EL*100% = 6.37%

n' n' = n = 30

vf vf from table = 1.286

un' un' from table = 1.168

kT kt = un'*vf = 1.503

TI TI = kt*SL = 6.22

TI% TI% = TI/EL*100% = 25.6%

Sy Sy = SQRT(Syy/(n-1)) = 7.266 

r2 r2 = 1-(SL^2/Sy^2) = 0.675 

r r = SQRT((1-SL^2/Sy^2)) = 0.822 

* Indicates correlation coefficient is undefined.

Plot of Logarithmic Regression C

y = 9.8067Ln(x) - 17.7

-5

05

10152025

0.0 5.0 10.0 15.0 20.0 25.0 

PM CEMS Response

PM Concentration

PS-11 Example 2-R 10-27-04.xls Logarithmic Summary 
CALCULATIONS FOR EXPONENTIAL CORRELATION

Facility: Facility B Emission Unit: IncineratorLocation: Anytown USA Test Dates: April 3 to 6, 2003

PM concentration

PM CEMS mg/acm

response Measured Transformed CALCULATED STATISTICAL PARAMETERS

Run x y y' = Ln(y) (xi-x~)2 (y'i-y'~)2 (xi-x~)(y'i-y'~) y'^ (y'^i-y'i)

1A 6.4 1.3 0.262 174.94 2.957 22.744 0.886 0.31B 6.4 1.2 0.182 174.94 3.239 23.803 0.886 0.42A 7.5 2.3 0.833 147.06 1.320 13.934 0.978 0.02B 7.5 2.5 0.916 147.06 1.136 12.923 0.978 0.03A 11.2 2.5 0.916 71.01 1.136 8.980 1.284 0.13B 11.2 2.8 1.030 71.01 0.907 8.025 1.284 0.04A 16.4 7.7 2.041 10.41 0.004 -0.191 1.715 0.14B 16.4 7.9 2.067 10.41 0.007 -0.274 1.715 0.15A 29.9 12.9 2.557 105.54 0.331 5.910 2.833 0.05B 29.9 12.5 2.526 105.54 0.296 5.587 2.833 0.06A 22.2 7.8 2.054 6.62 0.005 0.186 2.195 0.06B 22.2 7.3 1.988 6.62 0.000 0.015 2.195 0.07A 19.5 5.8 1.758 0.02 0.050 0.028 1.971 0.07B 19.5 5.9 1.775 0.02 0.043 0.026 1.971 0.08A 27.4 11.7 2.460 60.42 0.228 3.713 2.626 0.08B 27.4 12 2.485 60.42 0.253 3.910 2.626 0.09A 32.8 20.4 3.016 173.54 1.068 13.616 3.073 0.09B 32.8 20.9 3.040 173.54 1.119 13.935 3.073 0.0

10A 26.5 20.9 3.040 47.24 1.119 7.271 2.551 0.210B 26.5 21.3 3.059 47.24 1.159 7.401 2.551 0.211A 33.5 22.3 3.105 192.47 1.260 15.575 3.131 0.011B 33.5 22.6 3.118 192.47 1.291 15.760 3.131 0.012A 28.1 18.4 2.912 71.80 0.866 7.884 2.684 0.012B 28.1 17.8 2.879 71.80 0.805 7.603 2.684 0.013A 9.6 3.7 1.308 100.53 0.454 6.754 1.151 0.013B 9.6 3.9 1.361 100.53 0.386 6.226 1.151 0.014A 5.3 4.2 1.435 205.25 0.299 7.835 0.795 0.414B 5.3 4.3 1.459 205.25 0.274 7.497 0.795 0.415A 18.1 6.8 1.917 2.33 0.004 0.099 1.855 0.015B 18.1 7.1 1.960 2.33 0.000 0.033 1.855 0.0

PS-11 Example 2-R 10-27-04.xls Exponential Calcs
RESULTS OF EXPONENTIAL CORRELATION

Facility: Facility B Emission Unit: IncineratorLocation: Anytown USA Test Dates: April 3 to 6, 2003

Correlation equation: y = 1.428 e 0.0

Variable Equation Value

n Number of data points = 30 Summary of Acceptance Criteria for PS-11

x

~ x~ = 1/n*(Sum of (xi)) = 19.63 Criterion Actual Allowable 

Sxx Sxx = Sum((xi-x~)^2) = 2,738.4 Correlation coefficient 0.921 > 0.85 

y'~ y'~ = 1/n*(Sum of (y'i)) = 1.982 Confidence interval 3.80% < 10% 

Sy'y' Sy'y' = Sum((y'i-y'~)^2) = 22.02 Tolerance interval 15.9% < 25% 

Sxy' Sxy' = Sum((xi-x~)(y'i-y'~)) = 226.81 * Indicates correlation coefficient is undefined.

b0' b0' = y'~-b1x~ = 0.356

b0 b0 = e^b0' = 1.428

b1 b1 = Sxy'/Sxx = 0.083

SL SL = SQRT(1/(n-2)(Sum(y'i^-y'i)^2)) = 0.340

y'^mean y'^ at mean x value = 1.982

tf tf from table = 2.048

CI' CI' = tf*SL*SQRT(1/n) = 0.1270

LCL' LCL' = y~' - CI' 1.855

UCL' UCL' = y~' + CI' 2.109

CI CI = (e^UCL' - e^LCL')/2 0.924

EL Emission Limit = 24.3

CI% CI% =CI/EL*100% = 3.80%

n' n' = n = 30

vf vf from table = 1.286

un' un' from table = 1.168

kT kT = un'*vf = 1.503

TI' TI' = kt*SL = 0.510

LTL' LTL' = y~' - TI' 1.472

UTL' UTL' = y~' + TI' 2.492

TI TI = (e^UTL' - e^LTL')/2 3.866

TI% TI% = TI/EL*100% = 15.9%

Sy' Sy' = SQRT(Sy'y'/(n-1)) = 0.871 PS-11 Correlation 

r2 r2 = 1-(SL^2/Sy'^2) = 0.848 Calculation Spread

r r = SQRT((1-SL^2/Sy'^2)) = 0.921 Version 2-6 10/2

* Indicates correlation coefficient is undefined.

Plot of Exponential Regression Curve

y = 1.4281e0.0828x

05

10152025

0.0 5.0 10.0 15.0 20.0 25.0 30.0 

PM CEMS Response

PM Concentration

PS-11 Example 2-R 10-27-04.xls Exponential Summary 
CALCULATIONS FOR POWER CORRELATION

Facility: Facility B Emission Unit: IncineratorLocation: Anytown USA Test Dates: April 3 to 6, 2003

PM concentration

PM CEMS response mg/acm

Measured Transformed Measured Transformed CALCULATED STATISTICAL PARAMETERS

Run x x' = Ln(x) y y' = Ln(y) (x'i-x'~)2 (y'i-y'~)2 (x'i-x'~)(y'i-y'~) y'^ 

1A 6.4 1.856 1.3 0.262 0.929 2.957 1.658 0.756 1B 6.4 1.856 1.2 0.182 0.929 3.239 1.735 0.756 2A 7.5 2.015 2.3 0.833 0.649 1.320 0.925 0.957 2B 7.5 2.015 2.5 0.916 0.649 1.136 0.858 0.957 3A 11.2 2.416 2.5 0.916 0.163 1.136 0.431 1.468 3B 11.2 2.416 2.8 1.030 0.163 0.907 0.385 1.468 4A 16.4 2.797 7.7 2.041 0.001 0.004 -0.001 1.953 4B 16.4 2.797 7.9 2.067 0.001 0.007 -0.002 1.953 5A 29.9 3.398 12.9 2.557 0.334 0.331 0.332 2.717 5B 29.9 3.398 12.5 2.526 0.334 0.296 0.314 2.717 6A 22.2 3.100 7.8 2.054 0.078 0.005 0.020 2.338 6B 22.2 3.100 7.3 1.988 0.078 0.000 0.002 2.338 7A 19.5 2.970 5.8 1.758 0.023 0.050 -0.034 2.173 7B 19.5 2.970 5.9 1.775 0.023 0.043 -0.031 2.173 8A 27.4 3.311 11.7 2.460 0.240 0.228 0.234 2.606 8B 27.4 3.311 12 2.485 0.240 0.253 0.247 2.606 9A 32.8 3.490 20.4 3.016 0.449 1.068 0.693 2.835 9B 32.8 3.490 20.9 3.040 0.449 1.119 0.709 2.835 

10A 26.5 3.277 20.9 3.040 0.209 1.119 0.483 2.563 10B 26.5 3.277 21.3 3.059 0.209 1.159 0.492 2.563 11A 33.5 3.512 22.3 3.105 0.478 1.260 0.776 2.861 11B 33.5 3.512 22.6 3.118 0.478 1.291 0.785 2.861 12A 28.1 3.336 18.4 2.912 0.266 0.866 0.480 2.638 12B 28.1 3.336 17.8 2.879 0.266 0.805 0.463 2.638 13A 9.6 2.262 3.7 1.308 0.312 0.454 0.376 1.271 13B 9.6 2.262 3.9 1.361 0.312 0.386 0.347 1.271 14A 5.3 1.668 4.2 1.435 1.328 0.299 0.630 0.516 14B 5.3 1.668 4.3 1.459 1.328 0.274 0.603 0.516 15A 18.1 2.896 6.8 1.917 0.006 0.004 -0.005 2.078 15B 18.1 2.896 7.1 1.960 0.006 0.000 -0.002 2.078 

PS-11 Example 2-R 10-27-04.xls Power Calcs
RESULTS OF POWER CORRELATION

Facility: Facility B Emission Unit: IncineratorLocation: Anytown USA Test Dates: April 3 to 6, 2003

Correlation equation: y = 0.2007 x 1.2722

Variable Equation Value

n Number of data points = 30 Summary of Acceptance Criteria for PS-11

x'~ x'~ = 1/n*(Sum of (x'i)) = 2.820 Criterion Actual Allowable 

Sx'x' Sx'x' = Sum((x'i-x'~)^2) = 10.93 Correlation coefficient 0.892 > 0.85 

y'~ y'~ = 1/n*(Sum of (y'i)) = 1.982 Confidence interval 4.41% < 10% 

Sy'y' Sy'y' = Sum((y'i-y'~)^2) = 22.02 Tolerance interval 18.7% < 25% Sx'y' Sx'y' = Sum((x'i-x'~)(y'i-y'~)) = 13.90 * Indicates correlation coefficient is undefined.

b0' b0' = y'~-b1x~ = -1.606

b0 b0 = e^b0' = 0.201

b1 b1 = Sx'y'/Sx'x' = 1.272

SL SL = SQRT(1/(n-2)(Sum(y'i^-y'i)^2)) = 0.393

y'^mean y'^ at mean x value = 1.982

tf tf from table = 2.048

CI' CI = tf*SL*SQRT(1/n) = 0.147

LCL' LCL' = y~' - CI' 1.835

UCL' UCL' = y~' + CI' 2.129

CI CI = (e^UCL' - e^LCL')/2 1.071

EL Emission Limit = 24.3

CI% CI% =CI/EL*100% = 4.41%

n' n' = n = 30

vf vf from table = 1.286

un' un' from table = 1.168

kT kT = un'*vf = 1.503

TI' TI' = kTSL = 0.591

LTL' LTL' = y~' - TI' 1.391

UTL' UTL' = y~' + TI' 2.573

TI TI = (e^UTL' - e^LTL')/2 4.540

TI% TI% = TI/EL*100% = 18.7%

Sy' Sy' = SQRT(Sy'y'/(n-1)) = 0.871 PS-11 

r2 r2 = 1-(SL^2/Sy'^2) = 0.796 Calcula

r r = SQRT((1-SL^2/Sy'^2)) = 0.892 Version 

* Indicates correlation coefficient is undefined.

Plot of Power Function Regression Curve

y = 0.2007x1.2722

05

10152025

0.0 5.0 10.0 15.0 20.0 25.0 30.0 

PM CEMS Response

PM Concentration

PS-11 Example 2-R 10-27-04.xls Power Summary 
SELECTION OF BEST MODEL

Facility: Facility B Emission Unit: IncineratorLocation: Anytown USA Test Dates: April 3 to 6, 2

Model Correlation coefficient (a) > 0.85 ? Confidence interval half range percentage < 10% ? Tolerance interval half range percentage < 25% ? Min/max wit

allowable ran

Linear 0.895 Yes 4.98% Yes 20.0% Yes (b) 

Polynomial 0.919 Yes 5.91% Yes 18.0% Yes Yes 

Logarithmic 0.822 No 6.37% Yes 25.6% No (b) 

Exponential 0.921 Yes 3.80% Yes 15.9% Yes (b) 

Power 0.892 Yes 4.41% Yes 18.7% Yes (b) 

(a) * indicates correlation coefficient is undefined; model does not satisfy criterion.(b) Not applicable; criterion applies only to polynomial model.

Best m

PS-11 Correlation TestCalculation SpreadsheetVersion 2-6 10/25/04

PS-11 Example 2-R 10-27-04.xls Best Model 
Appendix A.3

Complete Results of Example No. 3 –

Initial Correlation Test
CORRELATION TEST PM CEMS AND REFERENCE METHOD TEST DATA

PM CEMS PM concentration Facility: Facility C

response mg/acm Location:

Anytown US

A

Run x y Emission Unit: HWI

1 5.8 3.5 Test Dates: November 1 to 4, 200

2

2 7.6 93 7.8 7.8

Value Units

4 16.4 25.2 Emission limit: 17.45 mg/acm5 10.1 16.66 11.1 14.37 12.0 19.68 13.2 20.19 15.7 23.4

10 17.6 26.711 21.7 26.812 21.0 28.913 22.3 25.714 24.1 26.215 24.7 26.2

05

101520253035

0.0 5.0 10.0 15.0 20.0 

PM CEMS Response

PM Concentration

PS-11 Example 3-R 10-27-04.xls Test Data
CALCULATIONS FOR LINEAR CORRELATION

Facility: Facility C Emission Unit: HWILocation: Anytown USA Test Dates: November 1 to 

PM CEMS PM concentration

response mg/acm CALCULATED STATISTICAL PARAMETER

Run x y (xi-x~)2 (yi-y~)2 (xi-x~)(yi-y~) y^ 

1 5.8 3.5 92.29 272.25 158.51 8.942 2 7.6 9 60.94 121.00 85.87 11.014 3 7.8 7.8 57.86 148.84 92.80 11.245 4 16.4 25.2 0.99 27.04 5.17 21.143 5 10.1 16.6 28.16 11.56 18.04 13.892 6 11.1 14.3 18.55 32.49 24.55 15.043 7 12 19.6 11.61 0.16 1.36 16.079 8 13.2 20.1 4.87 0.01 -0.22 17.460 9 15.7 23.4 0.09 11.56 1.00 20.338 

10 17.6 26.7 4.81 44.89 14.70 22.525 11 21.7 26.8 39.61 46.24 42.79 27.244 12 21 28.9 31.29 79.21 49.78 26.438 13 22.3 25.7 47.52 32.49 39.29 27.934 14 24.1 26.2 75.57 38.44 53.90 30.006 15 24.7 26.2 86.37 38.44 57.62 30.697 

PS-11 Example 3-R 10-27-04.xls Linear Calcs
RESULTS OF LINEAR CORRELATION

Facility: Facility C Emission Unit: HWILocation: Anytown USA Test Dates: November 1 to 4, 2002

Correlation equation: y = 2.267 

Variable Equation Value

n Number of data points = 15 Summary of Acceptance Criteria for PS-11

x

~ x~ = 1/n*(Sum of (xi)) = 15.41 Criterion Actual Allowable 

Sxx Sxx = Sum((xi-x~)^2) = 561 Correlation coefficient 0.898 > 0.85 

y

~ y~ = 1/n*(Sum of (yi)) = 20.00 Confidence interval 11.29% < 10% 

Syy Syy = Sum((yi-y~)^2) = 905 Tolerance interval 35.6% < 25% Sxy Sxy = Sum((xi-x~)(yi-y~)) = 645.2 * Indicates correlation coefficient is undefined.

b0 b0 = y~-b1x~ = 2.267

b1 b1 = Sxy/Sxx = 1.151

SL SL = SQRT(1/(n-2)(Sum(yi^-yi)^2)) = 3.530

y^mean y^ at mean x value = 20.000

tf tf from table = 2.160

CI CI = tf*SL*SQRT(1/n) = 1.969

EL Emission Limit = 17.5

CI% CI% =CI/EL*100% = 11.29%

n' n' = n = 15

vf vf from table = 1.485

un' un' from table = 1.184

kT kt = un'*vf = 1.759

TI TI = kt*SL = 6.210

TI% TI% = TI/EL*100% = 35.6%

Sy Sy = SQRT(Syy/(n-1)) = 8.038 PS-11 Correla

r2 r2 = 1-(SL^2/Sy^2) = 0.807 Calculation Sp

r r = SQRT((1-SL^2/Sy^2)) = 0.898 Version 2-6 

* Indicates correlation coefficient is undefined.

Plot of Linear Regression Line

y = 1.151x + 2.2665

05

101520253035

0.0 5.0 10.0 15.0 20.0 

PM CEMS Response

PM Concentration

PS-11 Example 3-R 10-27-04.xls Linear Summary 
CALCULATIONS FOR POLYNOMIAL CORRELATION

Facility: Facility C Emission Unit: HWILocation: Anytown USA Test Dates: November 1 to 4, 2002

PM CEMS PM concentration

response mg/acm CALCULATED STATISTICAL PARAMETERS

Run x y x2 x3 x4 xy x2y y^ (y^-y)2 delta (y

1 5.8 3.5 34 195 1,132 20 118 3.138 0.131 0.4679 

2 7.6 9 58 439 3,336 68 520 8.429 0.326 0.2223 

3 7.8 7.8 61 475 3,702 61 475 8.976 1.384 0.2060 

4 16.4 25.2 269 4,411 72,339 413 6,778 24.851 0.121 0.1650 

5 10.1 16.6 102 1,030 10,406 168 1,693 14.689 3.653 0.1214 

6 11.1 14.3 123 1,368 15,181 159 1,762 16.838 6.444 0.1224 

7 12 19.6 144 1,728 20,736 235 2,822 18.600 1.000 0.1320 

8 13.2 20.1 174 2,300 30,360 265 3,502 20.694 0.352 0.1488 

9 15.7 23.4 246 3,870 60,757 367 5,768 24.119 0.517 0.1672 

10 17.6 26.7 310 5,452 95,951 470 8,271 25.876 0.679 0.1541 

11 21.7 26.8 471 10,218 221,737 582 12,620 27.177 0.142 0.1374 

12 21 28.9 441 9,261 194,481 607 12,745 27.196 2.903 0.1265 

13 22.3 25.7 497 11,090 247,297 573 12,780 27.082 1.911 0.1565 

14 24.1 26.2 581 13,998 337,340 631 15,217 26.360 0.026 0.2949 

15 24.7 26.2 610 15,069 372,210 647 15,984 25.973 0.051 0.3775 

PS-11 Example 3-R 10-27-04.xls Polynomial Calcs
RESULTS OF POLYNOMIAL CORRELATION

Facility: Facility C Emission Unit: HWILocation: Anytown USA Test Dates: November 1 to 4, 2002

Variable Equation Value Correlation equation: y = -18.371 

n Number of data points = 15

S1 S1 = Sum (xi) = 231 Summary of Acceptance Criteria for PS-11

S2 S2 = Sum (xi2) = 4,121 Criterion Actual Allowable 

S3 S3 = Sum (xi3) = 80,903 Correlation coefficient 0.987 > 0.85 

S4 S4 = Sum (xi4) = 1,686,966 Confidence interval 5.56% < 10% 

S5 S5 = Sum (yi) = 300 Tolerance interval 13.4% < 25% S6 S6 = Sum (xiyi) = 5,267 * Indicates correlation coefficient is undefined.

S7 S7 = Sum (xi2yi) = 101,055

det A det A = nS2S4-S2S2S2+S1S3S2-S3S3n+S2S1S3-S4S1S1 = 1.168E+08 Check for Correlation Curve Minimum/Maximum

b0 b0 = (S5S2S4+S1S3S7+S2S6S3-S7S2S2-S3S3S5-S4S6S1)/det A = -18.371 Correlation curve minimum point 

b1 b1 = (nS6S4+S5S3S2+S2S1S7-S2S6S2-S7S3n-S4S1S5)/det A = 4.296 Minimum allowable x value 

b2 b2 = (nS2S7+S1S6S2+S5S1S3-S2S2S5-S3S6n-S7S1S1)/det A = -0.10123 Is correlation curve minimum < minimum x value? 

Sp Sp = Sqrt((1/(n-3)Sum of (y^-Y)2) = 1.279 Correlation curve maximum point 

D D = n(S2S4-S32)+S1(S3S2-S1S4)+S2(S1S3-S22) = 1.168E+08 Extrapolation limit for x (125% of maximum x value) 

C0 C0 = (S2S4-S32)/D = 3.481 Is correlation curve maximum > extrapolation limit? 

C1 C1 = (S3S2-S1S4)/D = -0.4833

C2 C2 = (S1S3-S22)/D = 1.467E-02

C3 C3 = (nS4-S22)/D = 7.124E-02C4 C4 = (S1S2-nS3)/D = -2.236E-03

C5 C5 = (nS2-S12)/D = 7.197E-05

tf tf from table = 2.179

EL Emission Limit = 17.5

CI CI = tf*Sp *sqrt(delta-min) = 0.971

CI% CI% = CI/EL*100= 5.56%

vf vf from table = 1.515

un' un' from table = 1.208

n' n' = 1/delta = 8.24

kT kt = un'*vf = 1.831

TI TI = kt*sp = 2.342

TI% TI% = TI/EL *100% = 13.4%

y~ y~ = 1/n*(Sum of (yi)) = 20.00

Sy Sy = sqrt(Sum of ( yi-y~)^2/(n-1) = 8.04

r2 r2 = 1-(Sp^2/Sy^2) = 0.975

r r = sqrt((1-Sp^2/Sy^2)) = 0.987

Max-min? b2 > 0? maximum 

xmax-min y = -b1/2b2 = 21.22 

1.25xmax 30.88 

* Indicates correlation coefficient is undefined.

Plot of Polynomial Regression C

y = -0.1012x2 + 4.2957x - 18.371

05

101520253035

0.0 5.0 10.0 15.0 

PM CEMS Respon

PM Concentration

PS-11 Example 3-R 10-27-04.xls Polynomial Summary 
CALCULATIONS FOR LOGARITHMIC CORRELATION

Facility: Facility C Emission Unit: HWILocation: Anytown USA Test Dates: November 1 to 4, 20

PM CEMS response PM concentration

Measured Transformed mg/acm CALCULATED STATISTICAL PARA

Run x x' = ln (x) y (xi'-x'~)2 (yi-y~)2 (x'i-x'~)(yi-y~) y^ 

1 5.8 1.758 3.5 0.785 272.250 14.620 52 7.6 2.028 9 0.379 121.000 6.774 93 7.8 2.054 7.8 0.348 148.840 7.196 104 16.4 2.797 25.2 0.024 27.040 0.797 225 10.1 2.313 16.6 0.110 11.560 1.127 146 11.1 2.407 14.3 0.056 32.490 1.351 167 12 2.485 19.6 0.025 0.160 0.064 178 13.2 2.580 20.1 0.004 0.010 -0.006 189 15.7 2.754 23.4 0.012 11.560 0.373 21

10 17.6 2.868 26.7 0.050 44.890 1.501 2311 21.7 3.077 26.8 0.188 46.240 2.947 2712 21 3.045 28.9 0.160 79.210 3.565 2613 22.3 3.105 25.7 0.212 32.490 2.626 2714 24.1 3.182 26.2 0.290 38.440 3.337 2915 24.7 3.207 26.2 0.317 38.440 3.490 29

PS-11 Example 3-R 10-27-04.xls Logarithmic Calcs
RESULTS OF LOGARITHMIC CORRELATION

Facility: Facility C Emission Unit: HWILocation: Anytown USA Test Dates: November 

Correlation equation: y = -24.443 + 

Variable Equation Value

n Number of data points = 15 Summary of Acceptance Criteria for PS-11

x'~ x'~ = 1/n*(Sum of (x'i)) = 2.64 Criterion Actual Allow

Sx'x' Sx'x' = Sum((x'i-x'~)^2) = 2.96 Correlation coefficient 0.959 > 0

y

~ y~ = 1/n*(Sum of (yi)) = 20.0 Confidence interval 7.32% < 1

Syy Syy = Sum((yi-y~)^2) = 905 Tolerance interval 23.1% < 25

Sx'y Sx'y = Sum((x'i-x'~)(yi-y~)) = 49.8 * Indicates correlation coefficient is undefined.

b0 b0 = y~-b1x'~ = -24.443

b1 b1 = Sx'y/Sx'x' = 16.809

SL SL = SQRT(1/(n-2)(Sum(yi^-yi)^2)) = 2.290

y^mean y^ at mean x value = 20.000

tf tf from table = 2.160

CI CI = tf*SL*SQRT(1/n) = 1.277

EL Emission Limit = 17.5

CI% CI% =CI/EL*100% = 7.32%

n' n' = n = 15

vf vf from table = 1.485

un' un' from table = 1.184

kT kt = un'*vf = 1.759

TI TI = kt*SL = 4.03

TI% TI% = TI/EL*100% = 23.1%

Sy Sy = SQRT(Syy/(n-1)) = 8.038 

r2 r2 = 1-(SL^2/Sy^2) = 0.919 

r r = SQRT((1-SL^2/Sy^2)) = 0.959 

* Indicates correlation coefficient is undefined.

Plot of Logarithmic Regression Cu

y = 16.809Ln(x) - 24.443

05

101520253035

0.0 5.0 10.0 15.0 20

PM CEMS Response

PM Concentration

PS-11 Example 3-R 10-27-04.xls Logarithmic Summary 
CALCULATIONS FOR EXPONENTIAL CORRELATION

Facility: Facility C Emission Unit: HWILocation: Anytown USA Test Dates: November 1 to 

PM concentration

PM CEMS mg/acm

response Measured Transformed CALCULATED STATISTICAL PARAMETERS

Run x y y' = Ln(y) (xi-x~)2 (y'i-y'~)2 (xi-x~)(y'i-y'~) y'^ (y'

1 5.8 3.5 1.253 92.29 2.614 15.533 2.115 2 7.6 9 2.197 60.94 0.452 5.249 2.256 3 7.8 7.8 2.054 57.86 0.665 6.203 2.272 4 16.4 25.2 3.227 0.99 0.128 0.355 2.948 5 10.1 16.6 2.809 28.16 0.004 0.320 2.453 6 11.1 14.3 2.660 18.55 0.044 0.902 2.531 7 12 19.6 2.976 11.61 0.011 -0.361 2.602 8 13.2 20.1 3.001 4.87 0.017 -0.289 2.696 9 15.7 23.4 3.153 0.09 0.080 0.083 2.893 

10 17.6 26.7 3.285 4.81 0.172 0.910 3.042 11 21.7 26.8 3.288 39.61 0.175 2.635 3.364 12 21 28.9 3.364 31.29 0.244 2.764 3.309 13 22.3 25.7 3.246 47.52 0.142 2.598 3.411 14 24.1 26.2 3.266 75.57 0.157 3.444 3.552 15 24.7 26.2 3.266 86.37 0.157 3.681 3.600 

PS-11 Example 3-R 10-27-04.xls Exponential Calcs
RESULTS OF EXPONENTIAL CORRELATION

Facility: Facility C Emission Unit: HWILocation: Anytown USA Test Dates: November 1 to 4, 2002

Correlation equation: y = 5.256 e 0.0

Variable Equation Value

n Number of data points = 15 Summary of Acceptance Criteria for PS-11

x

~ x~ = 1/n*(Sum of (xi)) = 15.41 Criterion Actual Allowable 

Sxx Sxx = Sum((xi-x~)^2) = 560.5 Correlation coefficient 0.812 > 0.85 

y'~ y'~ = 1/n*(Sum of (y'i)) = 2.870 Confidence interval 19.93% < 10% 

Sy'y' Sy'y' = Sum((y'i-y'~)^2) = 5.06 Tolerance interval 66.5% < 25% 

Sxy' Sxy' = Sum((xi-x~)(y'i-y'~)) = 44.03 * Indicates correlation coefficient is undefined.

b0' b0' = y'~-b1x~ = 1.659

b0 b0 = e^b0' = 5.256

b1 b1 = Sxy'/Sxx = 0.079

SL SL = SQRT(1/(n-2)(Sum(y'i^-y'i)^2)) = 0.351

y'^mean y'^ at mean x value = 2.870

tf tf from table = 2.160

CI' CI' = tf*SL*SQRT(1/n) = 0.1960

LCL' LCL' = y~' - CI' 2.674

UCL' UCL' = y~' + CI' 3.066

CI CI = (e^UCL' - e^LCL')/2 3.477

EL Emission Limit = 17.5

CI% CI% =CI/EL*100% = 19.93%

n' n' = n = 15

vf vf from table = 1.485

un' un' from table = 1.184

kT kT = un'*vf = 1.759

TI' TI' = kt*SL = 0.618

LTL' LTL' = y~' - TI' 2.252

UTL' UTL' = y~' + TI' 3.488

TI TI = (e^UTL' - e^LTL')/2 11.602

TI% TI% = TI/EL*100% = 66.5%

Sy' Sy' = SQRT(Sy'y'/(n-1)) = 0.601 PS-11 Correlation 

r2 r2 = 1-(SL^2/Sy'^2) = 0.659 Calculation Spread

r r = SQRT((1-SL^2/Sy'^2)) = 0.812 Version 2-6 10/2

* Indicates correlation coefficient is undefined.

Plot of Exponential Regression Curve

y = 5.2565e0.0785x

05

10152025303540

0.0 5.0 10.0 15.0 20.0 

PM CEMS Response

PM Concentration

PS-11 Example 3-R 10-27-04.xls Exponential Summary 
CALCULATIONS FOR POWER CORRELATION

Facility: Facility C Emission Unit: HWILocation: Anytown USA Test Dates: November 1 to 4, 

PM concentration

PM CEMS response mg/acm

Measured Transformed Measured Transformed CALCULATED STATISTICAL PARA

Run x x' = Ln(x) y y' = Ln(y) (x'i-x'~)2 (y'i-y'~)2 (x'i-x'~)(y'i-y'~) 

1 5.8 1.758 3.5 1.253 0.785 2.614 1.433 2 7.6 2.028 9 2.197 0.379 0.452 0.414 3 7.8 2.054 7.8 2.054 0.348 0.665 0.481 4 16.4 2.797 25.2 3.227 0.024 0.128 0.055 5 10.1 2.313 16.6 2.809 0.110 0.004 0.020 6 11.1 2.407 14.3 2.660 0.056 0.044 0.050 7 12 2.485 19.6 2.976 0.025 0.011 -0.017 8 13.2 2.580 20.1 3.001 0.004 0.017 -0.008 9 15.7 2.754 23.4 3.153 0.012 0.080 0.031 

10 17.6 2.868 26.7 3.285 0.050 0.172 0.093 11 21.7 3.077 26.8 3.288 0.188 0.175 0.181 12 21 3.045 28.9 3.364 0.160 0.244 0.198 13 22.3 3.105 25.7 3.246 0.212 0.142 0.174 14 24.1 3.182 26.2 3.266 0.290 0.157 0.213 15 24.7 3.207 26.2 3.266 0.317 0.157 0.223 

PS-11 Example 3-R 10-27-04.xls Power Calcs
RESULTS OF POWER CORRELATION

Facility: Facility C Emission Unit: HWILocation: Anytown USA Test Dates: November 1 to 4, 2002

Correlation equation: y = 0.7466 x 1.1959

Variable Equation Value

n Number of data points = 15 Summary of Acceptance Criteria for PS-11

x'~ x'~ = 1/n*(Sum of (x'i)) = 2.644 Criterion Actual Allowable 

Sx'x' Sx'x' = Sum((x'i-x'~)^2) = 2.96 Correlation coefficient 0.908 > 0.85 

y'~ y'~ = 1/n*(Sum of (y'i)) = 2.870 Confidence interval 14.28% < 10% 

Sy'y' Sy'y' = Sum((y'i-y'~)^2) = 5.06 Tolerance interval 46.4% < 25% Sx'y' Sx'y' = Sum((x'i-x'~)(y'i-y'~)) = 3.54 * Indicates correlation coefficient is undefined.

b0' b0' = y'~-b1x~ = -0.292

b0 b0 = e^b0' = 0.747

b1 b1 = Sx'y'/Sx'x' = 1.196

SL SL = SQRT(1/(n-2)(Sum(y'i^-y'i)^2)) = 0.253

y'^mean y'^ at mean x value = 2.870

tf tf from table = 2.160

CI' CI = tf*SL*SQRT(1/n) = 0.141

LCL' LCL' = y~' - CI' 2.729

UCL' UCL' = y~' + CI' 3.011

CI CI = (e^UCL' - e^LCL')/2 2.492

EL Emission Limit = 17.5

CI% CI% =CI/EL*100% = 14.28%

n' n' = n = 15

vf vf from table = 1.485

un' un' from table = 1.184

kT kT = un'*vf = 1.759

TI' TI' = kTSL = 0.444

LTL' LTL' = y~' - TI' 2.425

UTL' UTL' = y~' + TI' 3.314

TI TI = (e^UTL' - e^LTL')/2 8.093

TI% TI% = TI/EL*100% = 46.4%

Sy' Sy' = SQRT(Sy'y'/(n-1)) = 0.601 PS-11 

r2 r2 = 1-(SL^2/Sy'^2) = 0.824 Calcula

r r = SQRT((1-SL^2/Sy'^2)) = 0.908 Version 

* Indicates correlation coefficient is undefined.

Plot of Power Function Regression Curve

y = 0.7466x1.1959

05

10152025303540

0.0 5.0 10.0 15.0 20.0 

PM CEMS Response

PM Concentration

PS-11 Example 3-R 10-27-04.xls Power Summary 
SELECTION OF BEST MODEL

Facility: Facility C Emission Unit: HWILocation: Anytown USA Test Dates: November 1 t

Model Correlation coefficient (a) > 0.85 ? Confidence interval half range percentage < 10% ? Tolerance interval half range percentage < 25% ? Min/max wit

allowable ran

Linear 0.898 Yes 11.29% No 35.6% No (b) 

Polynomial 0.987 Yes 5.56% Yes 13.4% Yes No 

Logarithmic 0.959 Yes 7.32% Yes 23.1% Yes (b) 

Exponential 0.812 No 19.93% No 66.5% No (b) 

Power 0.908 Yes 14.28% No 46.4% No (b) 

(a) * indicates correlation coefficient is undefined; model does not satisfy criterion.(b) Not applicable; criterion applies only to polynomial model.

Best m

PS-11 Correlation TestCalculation SpreadsheetVersion 2-6 10/25/04

PS-11 Example 3-R 10-27-04.xls Best Model 
Appendix A.4

Complete Results of Example No. 4

Original Test Data
CORRELATION TEST PM CEMS AND REFERENCE METHOD TEST DATA

PM CEMS PM concentration Facility: Facility D

response mg/acm Location:

Anytown US

A

Run x y Emission Unit: Boiler

1 6.7 3.1 Test Dates: November 1 to 4, 200

2

2 7.6 7.43 7.8 2.1

Value Units

4 9.2 3.7 Emission limit: 18.5 mg/acm

5 10.1 6.16 11.1 2.87 12.0 4.98 18.2 8.39 13.7 4.5

10 14.6 8.211 19.7 13.012 18.0 11.113 20.3 11.914 20.2 10.615 19.7 9.2

02468

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PM CEMS Response

PM Concentration

PS-11 Example 4A-R 10-27-04.xls Test Data
CALCULATIONS FOR LINEAR CORRELATION

Facility: Facility D Emission Unit: BoilerLocation: Anytown USA Test Dates: November 1 to 

PM CEMS PM concentration

response mg/acm CALCULATED STATISTICAL PARAMETER

Run x y (xi-x~)2 (yi-y~)2 (xi-x~)(yi-y~) y^ 

1 6.7 3.1 52.22 16.21 29.10 2.808 2 7.6 7.4 40.03 0.07 -1.73 3.346 3 7.8 2.1 37.54 25.27 30.80 3.465 4 9.2 3.7 22.34 11.74 16.20 4.302 5 10.1 6.1 14.64 1.05 3.93 4.840 6 11.1 2.8 7.99 18.72 12.23 5.437 7 12 4.9 3.71 4.96 4.29 5.975 8 18.2 8.3 18.26 1.38 5.01 9.680 9 13.7 4.5 0.05 6.90 0.60 6.991 

10 14.6 8.2 0.45 1.15 0.72 7.529 11 19.7 13 33.33 34.50 33.91 10.577 12 18 11.1 16.59 15.79 16.18 9.561 13 20.3 11.9 40.62 22.78 30.42 10.935 14 20.2 10.6 39.35 12.06 21.79 10.876 15 19.7 9.2 33.33 4.30 11.97 10.577 

PS-11 Example 4A-R 10-27-04.xls Linear Calcs
RESULTS OF LINEAR CORRELATION

Facility: Facility D Emission Unit: BoilerLocation: Anytown USA Test Dates: November 1 to 4, 2002

Correlation equation: y = -1.196 

Variable Equation Value

n Number of data points = 15 Summary of Acceptance Criteria for PS-11

x

~ x~ = 1/n*(Sum of (xi)) = 13.93 Criterion Actual Allowable 

Sxx Sxx = Sum((xi-x~)^2) = 360 Correlation coefficient 0.841 > 0.85 

y

~ y~ = 1/n*(Sum of (yi)) = 7.13 Confidence interval 5.80% < 10% 

Syy Syy = Sum((yi-y~)^2) = 177 Tolerance interval 18.3% < 25% Sxy Sxy = Sum((xi-x~)(yi-y~)) = 215.4 * Indicates correlation coefficient is undefined.

b0 b0 = y~-b1x~ = -1.196

b1 b1 = Sxy/Sxx = 0.598

SL SL = SQRT(1/(n-2)(Sum(yi^-yi)^2)) = 1.925

y^mean y^ at mean x value = 7.127

tf tf from table = 2.160

CI CI = tf*SL*SQRT(1/n) = 1.074

EL Emission Limit = 18.5

CI% CI% =CI/EL*100% = 5.80%

n' n' = n = 15

vf vf from table = 1.485

un' un' from table = 1.184

kT kt = un'*vf = 1.759

TI TI = kt*SL = 3.386

TI% TI% = TI/EL*100% = 18.3%

Sy Sy = SQRT(Syy/(n-1)) = 3.555 PS-11 Correla

r2 r2 = 1-(SL^2/Sy^2) = 0.707 Calculation Sp

r r = SQRT((1-SL^2/Sy^2)) = 0.841 Version 2-6 

* Indicates correlation coefficient is undefined.

Plot of Linear Regression Line

y = 0.5976x - 1.196

02468

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PM CEMS Response

PM Concentration

PS-11 Example 4A-R 10-27-04.xls Linear Summary 
CALCULATIONS FOR POLYNOMIAL CORRELATION

Facility: Facility D Emission Unit: BoilerLocation: Anytown USA Test Dates: November 1 to 4, 2002

PM CEMS PM concentration

response mg/acm CALCULATED STATISTICAL PARAMETERS

Run x y x2 x3 x4 xy x2y y^ (y^-y)2 delta 

1 6.7 3.1 45 301 2,015 21 139 4.024 0.854 0.4111 

2 7.6 7.4 58 439 3,336 56 427 4.032 11.346 0.2412 

3 7.8 2.1 61 475 3,702 16 128 4.043 3.775 0.2158 

4 9.2 3.7 85 779 7,164 34 313 4.219 0.270 0.1296 

5 10.1 6.1 102 1,030 10,406 62 622 4.423 2.812 0.1307 

6 11.1 2.8 123 1,368 15,181 31 345 4.733 3.735 0.1559 

7 12 4.9 144 1,728 20,736 59 706 5.085 0.034 0.1838 

8 18.2 8.3 331 6,029 109,720 151 2,749 9.437 1.292 0.1253 

9 13.7 4.5 188 2,571 35,228 62 845 5.945 2.088 0.2145 

10 14.6 8.2 213 3,112 45,437 120 1,748 6.502 2.883 0.2103 

11 19.7 13 388 7,645 150,614 256 5,045 10.993 4.027 0.1825 

12 18 11.1 324 5,832 104,976 200 3,596 9.244 3.444 0.1262 

13 20.3 11.9 412 8,365 169,818 242 4,904 11.671 0.053 0.2523 

14 20.2 10.6 408 8,242 166,497 214 4,325 11.556 0.913 0.2382 

15 19.7 9.2 388 7,645 150,614 181 3,570 10.993 3.216 0.1825 

PS-11 Example 4A-R 10-27-04.xls Polynomial Calcs
RESULTS OF POLYNOMIAL CORRELATION

Facility: Facility D Emission Unit: BoilerLocation: Anytown USA Test Dates: November 1 to 4, 2002

Variable Equation Value Correlation equation: y = 6.189 

n Number of data points = 15

S1 S1 = Sum (xi) = 209 Summary of Acceptance Criteria for PS-11

S2 S2 = Sum (xi2) = 3,270 Criterion Actual Allowable 

S3 S3 = Sum (xi3) = 55,562 Correlation coefficient 0.855 > 0.85 

S4 S4 = Sum (xi4) = 995,443 Confidence interval 7.68% < 10% 

S5 S5 = Sum (yi) = 107 Tolerance interval 18.2% < 25% S6 S6 = Sum (xiyi) = 1,704 * Indicates correlation coefficient is undefined.

S7 S7 = Sum (xi2yi) = 29,463

det A det A = nS2S4-S2S2S2+S1S3S2-S3S3n+S2S1S3-S4S1S1 = 2.107E+07 Check for Correlation Curve Minimum/Maximum

b0 b0 = (S5S2S4+S1S3S7+S2S6S3-S7S2S2-S3S3S5-S4S6S1)/det A = 6.189 Correlation curve minimum point 

b1 b1 = (nS6S4+S5S3S2+S2S1S7-S2S6S2-S7S3n-S4S1S5)/det A = -0.615 Minimum allowable x value 

b2 b2 = (nS2S7+S1S6S2+S5S1S3-S2S2S5-S3S6n-S7S1S1)/det A = 0.04361 Is correlation curve minimum < minimum x value? 

Sp Sp = Sqrt((1/(n-3)Sum of (y^-Y)2) = 1.843 Correlation curve maximum point 

D D = n(S2S4-S32)+S1(S3S2-S1S4)+S2(S1S3-S22) = 2.107E+07 Extrapolation limit for x (125% of maximum x value) 

C0 C0 = (S2S4-S32)/D = 7.963 Is correlation curve maximum > extrapolation limit? 

C1 C1 = (S3S2-S1S4)/D = -1.2473

C2 C2 = (S1S3-S22)/D = 4.346E-02

C3 C3 = (nS4-S22)/D = 2.013E-01C4 C4 = (S1S2-nS3)/D = -7.138E-03

C5 C5 = (nS2-S12)/D = 2.567E-04

tf tf from table = 2.179

EL Emission Limit = 18.5

CI CI = tf*Sp *sqrt(delta-min) = 1.421

CI% CI% = CI/EL*100= 7.68%

vf vf from table = 1.515

un' un' from table = 1.208

n' n' = 1/delta = 7.98

kT kt = un'*vf = 1.831

TI TI = kt*sp = 3.373

TI% TI% = TI/EL *100% = 18.2%

y~ y~ = 1/n*(Sum of (yi)) = 7.13

Sy Sy = sqrt(Sum of ( yi-y~)^2/(n-1) = 3.55

r2 r2 = 1-(Sp^2/Sy^2) = 0.731

r r = sqrt((1-Sp^2/Sy^2)) = 0.855

Max-min? b2 > 0? minimum 

xmax-min y = -b1/2b2 = 7.05 

1.25xmax 25.38 

* Indicates correlation coefficient is undefined.

Plot of Polynomial Regression C

y = 0.0436x2 - 0.6153x + 6.189

02468

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PM CEMS Respon

PM Concentration

PS-11 Example 4A-R 10-27-04.xls Polynomial Summary 
CALCULATIONS FOR LOGARITHMIC CORRELATION

Facility: Facility D Emission Unit: BoilerLocation: Anytown USA Test Dates: November 1 to 4, 20

PM CEMS response PM concentration

Measured Transformed mg/acm CALCULATED STATISTICAL PARA

Run x x' = ln (x) y (xi'-x'~)2 (yi-y~)2 (x'i-x'~)(yi-y~) y^ 

1 6.7 1.902 3.1 0.439 16.214 2.669 22 7.6 2.028 7.4 0.288 0.075 -0.147 33 7.8 2.054 2.1 0.261 25.267 2.568 34 9.2 2.219 3.7 0.120 11.742 1.185 45 10.1 2.313 6.1 0.064 1.054 0.259 56 11.1 2.407 2.8 0.025 18.720 0.684 57 12 2.485 4.9 0.006 4.958 0.178 68 18.2 2.901 8.3 0.113 1.377 0.395 99 13.7 2.617 4.5 0.003 6.899 -0.138 7

10 14.6 2.681 8.2 0.013 1.152 0.124 711 19.7 2.981 13 0.173 34.496 2.441 1012 18 2.890 11.1 0.106 15.787 1.293 913 20.3 3.011 11.9 0.199 22.785 2.127 1014 20.2 3.006 10.6 0.194 12.064 1.530 1015 19.7 2.981 9.2 0.173 4.299 0.862 10

PS-11 Example 4A-R 10-27-04.xls Logarithmic Calcs
RESULTS OF LOGARITHMIC CORRELATION

Facility: Facility D Emission Unit: BoilerLocation: Anytown USA Test Dates: Novembe

Correlation equation: y = -11.764 

Variable Equation Value

n Number of data points = 15 Summary of Acceptance Criteria for PS-11

x'~ x'~ = 1/n*(Sum of (x'i)) = 2.57 Criterion Actual Allo

Sx'x' Sx'x' = Sum((x'i-x'~)^2) = 2.18 Correlation coefficient 0.801 > 0

y

~ y~ = 1/n*(Sum of (yi)) = 7.1 Confidence interval 6.41% < 

Syy Syy = Sum((yi-y~)^2) = 177 Tolerance interval 20.2% < 2

Sx'y Sx'y = Sum((x'i-x'~)(yi-y~)) = 16.0 * Indicates correlation coefficient is undefined.

b0 b0 = y~-b1x'~ = -11.764

b1 b1 = Sx'y/Sx'x' = 7.365

SL SL = SQRT(1/(n-2)(Sum(yi^-yi)^2)) = 2.127

y^mean y^ at mean x value = 7.127

tf tf from table = 2.160

CI CI = tf*SL*SQRT(1/n) = 1.187

EL Emission Limit = 18.5

CI% CI% =CI/EL*100% = 6.41%

n' n' = n = 15

vf vf from table = 1.485

un' un' from table = 1.184

kT kt = un'*vf = 1.759

TI TI = kt*SL = 3.74

TI% TI% = TI/EL*100% = 20.2%

Sy Sy = SQRT(Syy/(n-1)) = 3.555 

r2 r2 = 1-(SL^2/Sy^2) = 0.642 

r r = SQRT((1-SL^2/Sy^2)) = 0.801 

* Indicates correlation coefficient is undefined.

Plot of Logarithmic Regression C

y = 7.3647Ln(x) - 11.764

02468

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0.0 5.0 10.0 15.0 

PM CEMS Response

PM Concentration

PS-11 Example 4A-R 10-27-04.xls Logarithmic Summary 
CALCULATIONS FOR EXPONENTIAL CORRELATION

Facility: Facility D Emission Unit: BoilerLocation: Anytown USA Test Dates: November 1 to 

PM concentration

PM CEMS mg/acm

response Measured Transformed CALCULATED STATISTICAL PARAMETERS

Run x y y' = Ln(y) (xi-x~)2 (y'i-y'~)2 (xi-x~)(y'i-y'~) y'^ (y'

1 6.7 3.1 1.131 52.22 0.480 5.007 1.149 2 7.6 7.4 2.001 40.03 0.031 -1.121 1.233 3 7.8 2.1 0.742 37.54 1.171 6.631 1.252 4 9.2 3.7 1.308 22.34 0.266 2.438 1.383 5 10.1 6.1 1.808 14.64 0.000 0.061 1.467 6 11.1 2.8 1.030 7.99 0.631 2.246 1.560 7 12 4.9 1.589 3.71 0.055 0.453 1.644 8 18.2 8.3 2.116 18.26 0.085 1.248 2.223 9 13.7 4.5 1.504 0.05 0.102 0.073 1.803 

10 14.6 8.2 2.104 0.45 0.078 0.188 1.887 11 19.7 13 2.565 33.33 0.549 4.276 2.364 12 18 11.1 2.407 16.59 0.340 2.374 2.205 13 20.3 11.9 2.477 40.62 0.426 4.157 2.420 14 20.2 10.6 2.361 39.35 0.288 3.367 2.410 15 19.7 9.2 2.219 33.33 0.156 2.280 2.364 

PS-11 Example 4A-R 10-27-04.xls Exponential Calcs
RESULTS OF EXPONENTIAL CORRELATION

Facility: Facility D Emission Unit: BoilerLocation: Anytown USA Test Dates: November 1 to 4, 2002

Correlation equation: y = 1.687 e 0.0

Variable Equation Value

n Number of data points = 15 Summary of Acceptance Criteria for PS-11

x

~ x~ = 1/n*(Sum of (xi)) = 13.93 Criterion Actual Allowable 

Sxx Sxx = Sum((xi-x~)^2) = 360.5 Correlation coefficient 0.806 > 0.85 

y'~ y'~ = 1/n*(Sum of (y'i)) = 1.824 Confidence interval 6.41% < 10% 

Sy'y' Sy'y' = Sum((y'i-y'~)^2) = 4.66 Tolerance interval 21.3% < 25% 

Sxy' Sxy' = Sum((xi-x~)(y'i-y'~)) = 33.68 * Indicates correlation coefficient is undefined.

b0' b0' = y'~-b1x~ = 0.523

b0 b0 = e^b0' = 1.687

b1 b1 = Sxy'/Sxx = 0.093

SL SL = SQRT(1/(n-2)(Sum(y'i^-y'i)^2)) = 0.341

y'^mean y'^ at mean x value = 1.824

tf tf from table = 2.160

CI' CI' = tf*SL*SQRT(1/n) = 0.1903

LCL' LCL' = y~' - CI' 1.634

UCL' UCL' = y~' + CI' 2.015

CI CI = (e^UCL' - e^LCL')/2 1.187

EL Emission Limit = 18.5

CI% CI% =CI/EL*100% = 6.41%

n' n' = n = 15

vf vf from table = 1.485

un' un' from table = 1.184

kT kT = un'*vf = 1.759

TI' TI' = kt*SL = 0.600

LTL' LTL' = y~' - TI' 1.224

UTL' UTL' = y~' + TI' 2.424

TI TI = (e^UTL' - e^LTL')/2 3.947

TI% TI% = TI/EL*100% = 21.3%

Sy' Sy' = SQRT(Sy'y'/(n-1)) = 0.577 PS-11 Correlation 

r2 r2 = 1-(SL^2/Sy'^2) = 0.650 Calculation Spread

r r = SQRT((1-SL^2/Sy'^2)) = 0.806 Version 2-6 10/2

* Indicates correlation coefficient is undefined.

Plot of Exponential Regression Curve

y = 1.6872e0.0934x

02468

101214

0.0 5.0 10.0 15.0 2

PM CEMS Response

PM Concentration

PS-11 Example 4A-R 10-27-04.xls Exponential Summary 
CALCULATIONS FOR POWER CORRELATION

Facility: Facility D Emission Unit: BoilerLocation: Anytown USA Test Dates: November 1 to 4, 

PM concentration

PM CEMS response mg/acm

Measured Transformed Measured Transformed CALCULATED STATISTICAL PARA

Run x x' = Ln(x) y y' = Ln(y) (x'i-x'~)2 (y'i-y'~)2 (x'i-x'~)(y'i-y'~) 

1 6.7 1.902 3.1 1.131 0.439 0.480 0.459 2 7.6 2.028 7.4 2.001 0.288 0.031 -0.095 3 7.8 2.054 2.1 0.742 0.261 1.171 0.553 4 9.2 2.219 3.7 1.308 0.120 0.266 0.178 5 10.1 2.313 6.1 1.808 0.064 0.000 0.004 6 11.1 2.407 2.8 1.030 0.025 0.631 0.126 7 12 2.485 4.9 1.589 0.006 0.055 0.019 8 18.2 2.901 8.3 2.116 0.113 0.085 0.098 9 13.7 2.617 4.5 1.504 0.003 0.102 -0.017 

10 14.6 2.681 8.2 2.104 0.013 0.078 0.032 11 19.7 2.981 13 2.565 0.173 0.549 0.308 12 18 2.890 11.1 2.407 0.106 0.340 0.190 13 20.3 3.011 11.9 2.477 0.199 0.426 0.291 14 20.2 3.006 10.6 2.361 0.194 0.288 0.236 15 19.7 2.981 9.2 2.219 0.173 0.156 0.164 

PS-11 Example 4A-R 10-27-04.xls Power Calcs
RESULTS OF POWER CORRELATION

Facility: Facility D Emission Unit: BoilerLocation: Anytown USA Test Dates: November 1 to 4, 2002

Correlation equation: y = 0.3084 x 1.1698

Variable Equation Value

n Number of data points = 15 Summary of Acceptance Criteria for PS-11

x'~ x'~ = 1/n*(Sum of (x'i)) = 2.565 Criterion Actual Allowable 

Sx'x' Sx'x' = Sum((x'i-x'~)^2) = 2.18 Correlation coefficient 0.782 > 0.85 

y'~ y'~ = 1/n*(Sum of (y'i)) = 1.824 Confidence interval 6.76% < 10% 

Sy'y' Sy'y' = Sum((y'i-y'~)^2) = 4.66 Tolerance interval 22.6% < 25% Sx'y' Sx'y' = Sum((x'i-x'~)(y'i-y'~)) = 2.55 * Indicates correlation coefficient is undefined.

b0' b0' = y'~-b1x~ = -1.177

b0 b0 = e^b0' = 0.308

b1 b1 = Sx'y'/Sx'x' = 1.170

SL SL = SQRT(1/(n-2)(Sum(y'i^-y'i)^2)) = 0.360

y'^mean y'^ at mean x value = 1.824

tf tf from table = 2.160

CI' CI = tf*SL*SQRT(1/n) = 0.201

LCL' LCL' = y~' - CI' 1.624

UCL' UCL' = y~' + CI' 2.025

CI CI = (e^UCL' - e^LCL')/2 1.251

EL Emission Limit = 18.5

CI% CI% =CI/EL*100% = 6.76%

n' n' = n = 15

vf vf from table = 1.485

un' un' from table = 1.184

kT kT = un'*vf = 1.759

TI' TI' = kTSL = 0.632

LTL' LTL' = y~' - TI' 1.192

UTL' UTL' = y~' + TI' 2.457

TI TI = (e^UTL' - e^LTL')/2 4.187

TI% TI% = TI/EL*100% = 22.6%

Sy' Sy' = SQRT(Sy'y'/(n-1)) = 0.577 PS-11 

r2 r2 = 1-(SL^2/Sy'^2) = 0.612 Calcula

r r = SQRT((1-SL^2/Sy'^2)) = 0.782 Version 

* Indicates correlation coefficient is undefined.

Plot of Power Function Regression Curve

y = 0.3084x1.1698

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PM Concentration

PS-11 Example 4A-R 10-27-04.xls Power Summary 
SELECTION OF BEST MODEL

Facility: Facility D Emission Unit: BoilerLocation: Anytown USA Test Dates: November 1 to 

Model Correlation coefficient (a) > 0.85 ? Confidence interval half range percentage < 10% ? Tolerance interval half range percentage < 25% ? Min/max with

allowable rang

Linear 0.841 No 5.80% Yes 18.3% Yes (b) 

Polynomial 0.855 Yes 7.68% Yes 18.2% Yes No 

Logarithmic 0.801 No 6.41% Yes 20.2% Yes (b) 

Exponential 0.806 No 6.41% Yes 21.3% Yes (b) 

Power 0.782 No 6.76% Yes 22.6% Yes (b) 

(a) * indicates correlation coefficient is undefined; model does not satisfy criterion.(b) Not applicable; criterion applies only to polynomial model.

Best mod

PS-11 Correlation TestCalculation SpreadsheetVersion 2-6 10/25/04

PS-11 Example 4A-R 10-27-04.xls Best Model 
Appendix A.5

Complete Results of Example No. 4

With Five Additional Test Runs
CORRELATION TEST PM CEMS AND REFERENCE METHOD TEST DATA

PM CEMS PM concentration Facility: Facility D

response mg/acm Location:

Anytown US

A

Run x y Emission Unit: Boiler

1 6.7 3.1 Test Dates: 11/1-11/4/02, 1/7-1/8/0

3

2 7.6 7.43 7.8 2.1

Value Units

4 9.2 3.7 Emission limit: 18.5 mg/acm

5 10.1 6.16 11.1 2.87 12.0 4.98 18.2 8.39 13.7 4.5

10 14.6 8.211 19.7 13.012 18.0 11.113 20.3 11.914 20.2 10.615 19.7 9.216 5.2 2.417 8.6 4.818 14.1 6.219 14.5 7.020 17.3 8.6

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PM Concentration

PS-11 Example 4B-R 20 Runs 10-27-04.xls Test Data
CALCULATIONS FOR LINEAR CORRELATION

Facility: Facility D Emission Unit: BoilerLocation: Anytown USA Test Dates: 11/1-11/4/02, 1/

PM CEMS PM concentration

response mg/acm CALCULATED STATISTICAL PARAMETER

Run x y (xi-x~)2 (yi-y~)2 (xi-x~)(yi-y~) y^ 

1 6.7 3.1 45.29 13.65 24.87 2.936 2 7.6 7.4 33.99 0.37 -3.53 3.452 3 7.8 2.1 31.70 22.04 26.43 3.567 4 9.2 3.7 17.89 9.58 13.09 4.370 5 10.1 6.1 11.09 0.48 2.31 4.886 6 11.1 2.8 5.43 15.96 9.31 5.459 7 12 4.9 2.04 3.59 2.71 5.975 8 18.2 8.3 22.75 2.27 7.18 9.530 9 13.7 4.5 0.07 5.27 -0.62 6.950 

10 14.6 8.2 1.37 1.97 1.64 7.466 11 19.7 13 39.31 38.50 38.91 10.390 12 18 11.1 20.88 18.53 19.67 9.415 13 20.3 11.9 47.20 26.06 35.07 10.734 14 20.2 10.6 45.83 14.48 25.76 10.677 15 19.7 9.2 39.31 5.78 15.08 10.390 16 5.2 2.4 67.73 19.32 36.17 2.076 17 8.6 4.8 23.33 3.98 9.64 4.026 18 14.1 6.2 0.45 0.35 -0.40 7.179 19 14.5 7 1.14 0.04 0.22 7.408 20 17.3 8.6 14.98 3.26 6.99 9.014 

PS-11 Example 4B-R 20 Runs 10-27-04.xls Linear Calcs
RESULTS OF LINEAR CORRELATION

Facility: Facility D Emission Unit: BoilerLocation: Anytown USA Test Dates: 11/1-11/4/02, 1/7-1/8/03

Correlation equation: y = -0.905 

Variable Equation Value

n Number of data points = 20 Summary of Acceptance Criteria for PS-11

x

~ x~ = 1/n*(Sum of (xi)) = 13.43 Criterion Actual Allowable 

Sxx Sxx = Sum((xi-x~)^2) = 472 Correlation coefficient 0.861 > 0.85 

y

~ y~ = 1/n*(Sum of (yi)) = 6.80 Confidence interval 4.25% < 10% 

Syy Syy = Sum((yi-y~)^2) = 205 Tolerance interval 14.7% < 25% Sxy Sxy = Sum((xi-x~)(yi-y~)) = 270.5 * Indicates correlation coefficient is undefined.

b0 b0 = y~-b1x~ = -0.905

b1 b1 = Sxy/Sxx = 0.573

SL SL = SQRT(1/(n-2)(Sum(yi^-yi)^2)) = 1.673

y^mean y^ at mean x value = 6.795

tf tf from table = 2.101

CI CI = tf*SL*SQRT(1/n) = 0.786

EL Emission Limit = 18.5

CI% CI% =CI/EL*100% = 4.25%

n' n' = n = 20

vf vf from table = 1.384

un' un' from table = 1.177

kT kt = un'*vf = 1.629

TI TI = kt*SL = 2.726

TI% TI% = TI/EL*100% = 14.7%

Sy Sy = SQRT(Syy/(n-1)) = 3.289 PS-11 Correla

r2 r2 = 1-(SL^2/Sy^2) = 0.741 Calculation Sp

r r = SQRT((1-SL^2/Sy^2)) = 0.861 Version 2-6 

* Indicates correlation coefficient is undefined.

Plot of Linear Regression Line

y = 0.5733x - 0.905

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PM Concentration

PS-11 Example 4B-R 20 Runs 10-27-04.xls Linear Summary 
CALCULATIONS FOR POLYNOMIAL CORRELATION

Facility: Facility D Emission Unit: BoilerLocation: Anytown USA Test Dates: 11/1-11/4/02, 1/7-1/8/03

PM CEMS PM concentration

response mg/acm CALCULATED STATISTICAL PARAMETERS

Run x y x2 x3 x4 xy x2y y^ (y^-y)2 delta 

1 6.7 3.1 45 301 2,015 21 139 3.578 0.229 0.2035 

2 7.6 7.4 58 439 3,336 56 427 3.745 13.360 0.1340 

3 7.8 2.1 61 475 3,702 16 128 3.789 2.852 0.1240 

4 9.2 3.7 85 779 7,164 34 313 4.167 0.218 0.0937 

5 10.1 6.1 102 1,030 10,406 62 622 4.474 2.643 0.0972 

6 11.1 2.8 123 1,368 15,181 31 345 4.876 4.310 0.1090 

7 12 4.9 144 1,728 20,736 59 706 5.291 0.153 0.1197 

8 18.2 8.3 331 6,029 109,720 151 2,749 9.531 1.516 0.0982 

9 13.7 4.5 188 2,571 35,228 62 845 6.214 2.937 0.1259 

10 14.6 8.2 213 3,112 45,437 120 1,748 6.776 2.029 0.1195 

11 19.7 13 388 7,645 150,614 256 5,045 10.919 4.329 0.1725 

12 18 11.1 324 5,832 104,976 200 3,596 9.357 3.039 0.0947 

13 20.3 11.9 412 8,365 169,818 242 4,904 11.514 0.149 0.2351 

14 20.2 10.6 408 8,242 166,497 214 4,325 11.413 0.662 0.2230 

15 19.7 9.2 388 7,645 150,614 181 3,570 10.919 2.956 0.1725 

16 5.2 2.4 27 141 731 12 65 3.413 1.027 0.4433 

17 8.6 4.8 74 636 5,470 41 355 3.990 0.657 0.0996 

18 14.1 6.2 199 2,803 39,525 87 1,233 6.457 0.066 0.1238 

19 14.5 7 210 3,049 44,205 102 1,472 6.711 0.084 0.1205 

20 17.3 8.6 299 5,178 89,575 149 2,574 8.766 0.028 0.0903 

PS-11 Example 4B-R 20 Runs 10-27-04.xls Polynomial Calcs
RESULTS OF POLYNOMIAL CORRELATION

Facility: Facility D Emission Unit: BoilerLocation: Anytown USA Test Dates: 11/1-11/4/02, 1/7-1/8/03

Variable Equation Value Correlation equation: y = 3.935 

n Number of data points = 20

S1 S1 = Sum (xi) = 269 Summary of Acceptance Criteria for PS-11

S2 S2 = Sum (xi2) = 4,079 Criterion Actual Allowable 

S3 S3 = Sum (xi3) = 67,368 Correlation coefficient 0.875 > 0.85 

S4 S4 = Sum (xi4) = 1,174,949 Confidence interval 5.47% < 10% 

S5 S5 = Sum (yi) = 136 Tolerance interval 14.4% < 25% S6 S6 = Sum (xiyi) = 2,096 * Indicates correlation coefficient is undefined.

S7 S7 = Sum (xi2yi) = 35,161

det A det A = nS2S4-S2S2S2+S1S3S2-S3S3n+S2S1S3-S4S1S1 = 6.865E+07 Check for Correlation Curve Minimum/Maximum

b0 b0 = (S5S2S4+S1S3S7+S2S6S3-S7S2S2-S3S3S5-S4S6S1)/det A = 3.935 Correlation curve minimum point 

b1 b1 = (nS6S4+S5S3S2+S2S1S7-S2S6S2-S7S3n-S4S1S5)/det A = -0.263 Minimum allowable x value 

b2 b2 = (nS2S7+S1S6S2+S5S1S3-S2S2S5-S3S6n-S7S1S1)/det A = 0.03137 Is correlation curve minimum < minimum x value? 

Sp Sp = Sqrt((1/(n-3)Sum of (y^-Y)2) = 1.595 Correlation curve maximum point 

D D = n(S2S4-S32)+S1(S3S2-S1S4)+S2(S1S3-S22) = 6.865E+07 Extrapolation limit for x (125% of maximum x value) 

C0 C0 = (S2S4-S32)/D = 3.705 Is correlation curve maximum > extrapolation limit? 

C1 C1 = (S3S2-S1S4)/D = -0.5942

C2 C2 = (S1S3-S22)/D = 2.121E-02

C3 C3 = (nS4-S22)/D = 9.993E-02C4 C4 = (S1S2-nS3)/D = -3.667E-03

C5 C5 = (nS2-S12)/D = 1.375E-04

tf tf from table = 2.110

EL Emission Limit = 18.5

CI CI = tf*Sp *sqrt(delta-min) = 1.011

CI% CI% = CI/EL*100= 5.47%

vf vf from table = 1.400

un' un' from table = 1.195

n' n' = 1/delta = 11.07

kT kt = un'*vf = 1.673

TI TI = kt*sp = 2.668

TI% TI% = TI/EL *100% = 14.4%

y~ y~ = 1/n*(Sum of (yi)) = 6.80

Sy Sy = sqrt(Sum of ( yi-y~)^2/(n-1) = 3.29

r2 r2 = 1-(Sp^2/Sy^2) = 0.765

r r = sqrt((1-Sp^2/Sy^2)) = 0.875

Max-min? b2 > 0? minimum 

xmax-min y = -b1/2b2 = 4.20 

1.25xmax 25.38 

* Indicates correlation coefficient is undefined.

Plot of Polynomial Regression C

y = 0.0314x2 - 0.2634x + 3.9349

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PM Concentration

PS-11 Example 4B-R 20 Runs 10-27-04.xls Polynomial Summary 
CALCULATIONS FOR LOGARITHMIC CORRELATION

Facility: Facility D Emission Unit: BoilerLocation: Anytown USA Test Dates: 11/1-11/4/02, 1/7-1/8/

PM CEMS response PM concentration

Measured Transformed mg/acm CALCULATED STATISTICAL PARA

Run x x' = ln (x) y (xi'-x'~)2 (yi-y~)2 (x'i-x'~)(yi-y~) y^ 

1 6.7 1.902 3.1 0.385 13.653 2.292 22 7.6 2.028 7.4 0.244 0.366 -0.299 33 7.8 2.054 2.1 0.219 22.043 2.198 34 9.2 2.219 3.7 0.092 9.579 0.938 45 10.1 2.313 6.1 0.044 0.483 0.146 56 11.1 2.407 2.8 0.013 15.960 0.461 67 12 2.485 4.9 0.001 3.591 0.071 68 18.2 2.901 8.3 0.144 2.265 0.570 99 13.7 2.617 4.5 0.009 5.267 -0.218 7

10 14.6 2.681 8.2 0.025 1.974 0.223 711 19.7 2.981 13 0.210 38.502 2.844 912 18 2.890 11.1 0.135 18.533 1.584 913 20.3 3.011 11.9 0.238 26.061 2.493 1014 20.2 3.006 10.6 0.234 14.478 1.839 915 19.7 2.981 9.2 0.210 5.784 1.102 916 5.2 1.649 2.4 0.763 19.316 3.840 117 8.6 2.152 4.8 0.137 3.980 0.739 418 14.1 2.646 6.2 0.015 0.354 -0.074 719 14.5 2.674 7 0.023 0.042 0.031 720 17.3 2.851 8.6 0.108 3.258 0.593 8

PS-11 Example 4B-R 20 Runs 10-27-04.xls Logarithmic Calcs
RESULTS OF LOGARITHMIC CORRELATION

Facility: Facility D Emission Unit: BoilerLocation: Anytown USA Test Dates: 11/1-11/4/

Correlation equation: y = -9.788 

Variable Equation Value

n Number of data points = 20 Summary of Acceptance Criteria for PS-11

x'~ x'~ = 1/n*(Sum of (x'i)) = 2.52 Criterion Actual Allo

Sx'x' Sx'x' = Sum((x'i-x'~)^2) = 3.25 Correlation coefficient 0.816 > 0

y

~ y~ = 1/n*(Sum of (yi)) = 6.8 Confidence interval 4.82% < 

Syy Syy = Sum((yi-y~)^2) = 205 Tolerance interval 16.7% < 2

Sx'y Sx'y = Sum((x'i-x'~)(yi-y~)) = 21.4 * Indicates correlation coefficient is undefined.

b0 b0 = y~-b1x'~ = -9.788

b1 b1 = Sx'y/Sx'x' = 6.574

SL SL = SQRT(1/(n-2)(Sum(yi^-yi)^2)) = 1.900

y^mean y^ at mean x value = 6.795

tf tf from table = 2.101

CI CI = tf*SL*SQRT(1/n) = 0.893

EL Emission Limit = 18.5

CI% CI% =CI/EL*100% = 4.82%

n' n' = n = 20

vf vf from table = 1.384

un' un' from table = 1.177

kT kt = un'*vf = 1.629

TI TI = kt*SL = 3.09

TI% TI% = TI/EL*100% = 16.7%

Sy Sy = SQRT(Syy/(n-1)) = 3.289 

r2 r2 = 1-(SL^2/Sy^2) = 0.666 

r r = SQRT((1-SL^2/Sy^2)) = 0.816 

* Indicates correlation coefficient is undefined.

Plot of Logarithmic Regression C

y = 6.5743Ln(x) - 9.7878

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PM Concentration

PS-11 Example 4B-R 20 Runs 10-27-04.xls Logarithmic Summary 
CALCULATIONS FOR EXPONENTIAL CORRELATION

Facility: Facility D Emission Unit: BoilerLocation: Anytown USA Test Dates: 11/1-11/4/0

PM concentration

PM CEMS mg/acm

response Measured Transformed CALCULATED STATISTICAL PARAMETE

Run x y y' = Ln(y) (xi-x~)2 (y'i-y'~)2 (xi-x~)(y'i-y'~) y'^ 

1 6.7 3.1 1.131 45.29 0.429 4.409 1.157 2 7.6 7.4 2.001 33.99 0.046 -1.253 1.241 3 7.8 2.1 0.742 31.70 1.091 5.881 1.260 4 9.2 3.7 1.308 17.89 0.229 2.023 1.391 5 10.1 6.1 1.808 11.09 0.000 -0.073 1.475 6 11.1 2.8 1.030 5.43 0.573 1.763 1.569 7 12 4.9 1.589 2.04 0.039 0.282 1.653 8 18.2 8.3 2.116 22.75 0.109 1.573 2.233 9 13.7 4.5 1.504 0.07 0.080 -0.076 1.812 

10 14.6 8.2 2.104 1.37 0.101 0.372 1.896 11 19.7 13 2.565 39.31 0.606 4.881 2.373 12 18 11.1 2.407 20.88 0.385 2.836 2.214 13 20.3 11.9 2.477 47.20 0.476 4.741 2.429 14 20.2 10.6 2.361 45.83 0.330 3.889 2.420 15 19.7 9.2 2.219 39.31 0.187 2.713 2.373 16 5.2 2.4 0.875 67.73 0.830 7.498 1.017 17 8.6 4.8 1.569 23.33 0.047 1.052 1.335 18 14.1 6.2 1.825 0.45 0.001 0.026 1.849 19 14.5 7 1.946 1.14 0.025 0.171 1.887 20 17.3 8.6 2.152 14.98 0.133 1.414 2.148 

PS-11 Example 4B-R 20 Runs 10-27-04.xls Exponential Calcs
RESULTS OF EXPONENTIAL CORRELATION

Facility: Facility D Emission Unit: BoilerLocation: Anytown USA Test Dates: 11/1-11/4/02, 1/7-1/8/03

Correlation equation: y = 1.700 e 0.0

Variable Equation Value

n Number of data points = 20 Summary of Acceptance Criteria for PS-11

x

~ x~ = 1/n*(Sum of (xi)) = 13.43 Criterion Actual Allowable 

Sxx Sxx = Sum((xi-x~)^2) = 471.8 Correlation coefficient 0.840 > 0.85 

y'~ y'~ = 1/n*(Sum of (y'i)) = 1.786 Confidence interval 4.52% < 10% 

Sy'y' Sy'y' = Sum((y'i-y'~)^2) = 5.72 Tolerance interval 16.3% < 25% 

Sxy' Sxy' = Sum((xi-x~)(y'i-y'~)) = 44.12 * Indicates correlation coefficient is undefined.

b0' b0' = y'~-b1x~ = 0.531

b0 b0 = e^b0' = 1.700

b1 b1 = Sxy'/Sxx = 0.094

SL SL = SQRT(1/(n-2)(Sum(y'i^-y'i)^2)) = 0.298

y'^mean y'^ at mean x value = 1.786

tf tf from table = 2.101

CI' CI' = tf*SL*SQRT(1/n) = 0.1398

LCL' LCL' = y~' - CI' 1.647

UCL' UCL' = y~' + CI' 1.926

CI CI = (e^UCL' - e^LCL')/2 0.837

EL Emission Limit = 18.5

CI% CI% =CI/EL*100% = 4.52%

n' n' = n = 20

vf vf from table = 1.384

un' un' from table = 1.177

kT kT = un'*vf = 1.629

TI' TI' = kt*SL = 0.485

LTL' LTL' = y~' - TI' 1.302

UTL' UTL' = y~' + TI' 2.271

TI TI = (e^UTL' - e^LTL')/2 3.007

TI% TI% = TI/EL*100% = 16.3%

Sy' Sy' = SQRT(Sy'y'/(n-1)) = 0.549 PS-11 Correlation 

r2 r2 = 1-(SL^2/Sy'^2) = 0.706 Calculation Spread

r r = SQRT((1-SL^2/Sy'^2)) = 0.840 Version 2-6 10/2

* Indicates correlation coefficient is undefined.

Plot of Exponential Regression Curve

y = 1.7e0.0935x

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PM CEMS Response

PM Concentration

PS-11 Example 4B-R 20 Runs 10-27-04.xls Exponential Summary 
CALCULATIONS FOR POWER CORRELATION

Facility: Facility D Emission Unit: BoilerLocation: Anytown USA Test Dates: 11/1-11/4/02, 1/7-

PM concentration

PM CEMS response mg/acm

Measured Transformed Measured Transformed CALCULATED STATISTICAL PARA

Run x x' = Ln(x) y y' = Ln(y) (x'i-x'~)2 (y'i-y'~)2 (x'i-x'~)(y'i-y'~) 

1 6.7 1.902 3.1 1.131 0.385 0.429 0.406 2 7.6 2.028 7.4 2.001 0.244 0.046 -0.106 3 7.8 2.054 2.1 0.742 0.219 1.091 0.489 4 9.2 2.219 3.7 1.308 0.092 0.229 0.145 5 10.1 2.313 6.1 1.808 0.044 0.000 -0.005 6 11.1 2.407 2.8 1.030 0.013 0.573 0.087 7 12 2.485 4.9 1.589 0.001 0.039 0.007 8 18.2 2.901 8.3 2.116 0.144 0.109 0.125 9 13.7 2.617 4.5 1.504 0.009 0.080 -0.027 

10 14.6 2.681 8.2 2.104 0.025 0.101 0.050 11 19.7 2.981 13 2.565 0.210 0.606 0.357 12 18 2.890 11.1 2.407 0.135 0.385 0.228 13 20.3 3.011 11.9 2.477 0.238 0.476 0.337 14 20.2 3.006 10.6 2.361 0.234 0.330 0.278 15 19.7 2.981 9.2 2.219 0.210 0.187 0.198 16 5.2 1.649 2.4 0.875 0.763 0.830 0.796 17 8.6 2.152 4.8 1.569 0.137 0.047 0.081 18 14.1 2.646 6.2 1.825 0.015 0.001 0.005 19 14.5 2.674 7 1.946 0.023 0.025 0.024 20 17.3 2.851 8.6 2.152 0.108 0.133 0.120 

PS-11 Example 4B-R 20 Runs 10-27-04.xls Power Calcs
RESULTS OF POWER CORRELATION

Facility: Facility D Emission Unit: BoilerLocation: Anytown USA Test Dates: 11/1-11/4/02, 1/7-1/8/03

Correlation equation: y = 0.3665 x 1.1062

Variable Equation Value

n Number of data points = 20 Summary of Acceptance Criteria for PS-11

x'~ x'~ = 1/n*(Sum of (x'i)) = 2.522 Criterion Actual Allowable 

Sx'x' Sx'x' = Sum((x'i-x'~)^2) = 3.25 Correlation coefficient 0.824 > 0.85 

y'~ y'~ = 1/n*(Sum of (y'i)) = 1.786 Confidence interval 4.73% < 10% 

Sy'y' Sy'y' = Sum((y'i-y'~)^2) = 5.72 Tolerance interval 17.0% < 25% Sx'y' Sx'y' = Sum((x'i-x'~)(y'i-y'~)) = 3.60 * Indicates correlation coefficient is undefined.

b0' b0' = y'~-b1x~ = -1.004

b0 b0 = e^b0' = 0.367

b1 b1 = Sx'y'/Sx'x' = 1.106

SL SL = SQRT(1/(n-2)(Sum(y'i^-y'i)^2)) = 0.311

y'^mean y'^ at mean x value = 1.786

tf tf from table = 2.101

CI' CI = tf*SL*SQRT(1/n) = 0.146

LCL' LCL' = y~' - CI' 1.640

UCL' UCL' = y~' + CI' 1.933

CI CI = (e^UCL' - e^LCL')/2 0.875

EL Emission Limit = 18.5

CI% CI% =CI/EL*100% = 4.73%

n' n' = n = 20

vf vf from table = 1.384

un' un' from table = 1.177

kT kT = un'*vf = 1.629

TI' TI' = kTSL = 0.507

LTL' LTL' = y~' - TI' 1.280

UTL' UTL' = y~' + TI' 2.293

TI TI = (e^UTL' - e^LTL')/2 3.154

TI% TI% = TI/EL*100% = 17.0%

Sy' Sy' = SQRT(Sy'y'/(n-1)) = 0.549 PS-11 

r2 r2 = 1-(SL^2/Sy'^2) = 0.679 Calcula

r r = SQRT((1-SL^2/Sy'^2)) = 0.824 Version 

* Indicates correlation coefficient is undefined.

Plot of Power Function Regression Curve

y = 0.3665x1.1062

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PM CEMS Response

PM Concentration

PS-11 Example 4B-R 20 Runs 10-27-04.xls Power Summary 
SELECTION OF BEST MODEL

Facility: Facility D Emission Unit: BoilerLocation: Anytown USA Test Dates: 11/1-11/4/02, 1

Model Correlation coefficient (a) > 0.85 ? Confidence interval half range percentage < 10% ? Tolerance interval half range percentage < 25% ? Min/max wit

allowable ran

Linear 0.861 Yes 4.25% Yes 14.7% Yes (b) 

Polynomial 0.875 Yes 5.47% Yes 14.4% Yes Yes 

Logarithmic 0.816 No 4.82% Yes 16.7% Yes (b) 

Exponential 0.840 No 4.52% Yes 16.3% Yes (b) 

Power 0.824 No 4.73% Yes 17.0% Yes (b) 

(a) * indicates correlation coefficient is undefined; model does not satisfy criterion.(b) Not applicable; criterion applies only to polynomial model.

Best m

PS-11 Correlation TestCalculation SpreadsheetVersion 2-6 10/25/04

PS-11 Example 4B-R 20 Runs 10-27-04.xls Best Model 
Appendix A.6

Complete Results of Example No. 4 –

With Run 2 Excluded
CORRELATION TEST PM CEMS AND REFERENCE METHOD TEST DATA

PM CEMS PM concentration Facility: Facility D

response mg/acm Location:

Anytown US

A

Run x y Emission Unit: Boiler

1 6.7 3.1 Test Dates: 11/1-11/4/02, 1/7-1/8/0

3

3 7.8 2.14 9.2 3.7

Value Units

5 10.1 6.1 Emission limit: 18.5 mg/acm

6 11.1 2.87 12.0 4.98 18.2 8.39 13.7 4.5

10 14.6 8.211 19.7 1312 18.0 11.113 20.3 11.914 20.2 10.615 19.7 9.216 5.2 2.417 8.6 4.818 14.1 6.219 14.5 720 17.3 8.6

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PM CEMS Response

PM Concentration

PS-11 Example 4C-R 19 Runs 10-27-04.xls Test Data
CALCULATIONS FOR LINEAR CORRELATION

Facility: Facility D Emission Unit: BoilerLocation: Anytown USA Test Dates: 11/1-11/4/02, 1/

PM CEMS PM concentration

response mg/acm CALCULATED STATISTICAL PARAMETER

Run x y (xi-x~)2 (yi-y~)2 (xi-x~)(yi-y~) y^ 

1 6.7 3.1 49.52 13.42 25.78 2.338 3 7.8 2.1 35.25 21.75 27.68 3.029 4 9.2 3.7 20.58 9.38 13.90 3.910 5 10.1 6.1 13.23 0.44 2.41 4.476 6 11.1 2.8 6.95 15.71 10.45 5.105 7 12 4.9 3.02 3.47 3.24 5.671 8 18.2 8.3 19.92 2.36 6.86 9.570 9 13.7 4.5 0.00 5.12 0.08 6.740 

10 14.6 8.2 0.75 2.06 1.24 7.306 11 19.7 13 35.56 38.90 37.19 10.513 12 18 11.1 18.17 18.81 18.49 9.444 13 20.3 11.9 43.08 26.39 33.71 10.891 14 20.2 10.6 41.77 14.72 24.80 10.828 15 19.7 9.2 35.56 5.94 14.53 10.513 16 5.2 2.4 72.88 19.04 37.25 1.394 17 8.6 4.8 26.39 3.85 10.08 3.533 18 14.1 6.2 0.13 0.32 -0.20 6.992 19 14.5 7 0.58 0.06 0.18 7.243 20 17.3 8.6 12.70 3.37 6.54 9.004 

PS-11 Example 4C-R 19 Runs 10-27-04.xls Linear Calcs
RESULTS OF LINEAR CORRELATION

Facility: Facility D Emission Unit: BoilerLocation: Anytown USA Test Dates: 11/1-11/4/02, 1/7-1/8/03

Correlation equation: y = -1.876 

Variable Equation Value

n Number of data points = 19 Summary of Acceptance Criteria for PS-11

x

~ x~ = 1/n*(Sum of (xi)) = 13.74 Criterion Actual Allowable 

Sxx Sxx = Sum((xi-x~)^2) = 436 Correlation coefficient 0.912 > 0.85 

y

~ y~ = 1/n*(Sum of (yi)) = 6.76 Confidence interval 3.63% < 10% 

Syy Syy = Sum((yi-y~)^2) = 205 Tolerance interval 12.4% < 25% Sxy Sxy = Sum((xi-x~)(yi-y~)) = 274.2 * Indicates correlation coefficient is undefined.

b0 b0 = y~-b1x~ = -1.876

b1 b1 = Sxy/Sxx = 0.629

SL SL = SQRT(1/(n-2)(Sum(yi^-yi)^2)) = 1.386

y^mean y^ at mean x value = 6.763

tf tf from table = 2.110

CI CI = tf*SL*SQRT(1/n) = 0.671

EL Emission Limit = 18.5

CI% CI% =CI/EL*100% = 3.63%

n' n' = n = 19

vf vf from table = 1.400

un' un' from table = 1.178

kT kt = un'*vf = 1.649

TI TI = kt*SL = 2.285

TI% TI% = TI/EL*100% = 12.4%

Sy Sy = SQRT(Syy/(n-1)) = 3.376 PS-11 Correla

r2 r2 = 1-(SL^2/Sy^2) = 0.831 Calculation Sp

r r = SQRT((1-SL^2/Sy^2)) = 0.912 Version 2-6 

* Indicates correlation coefficient is undefined.

Plot of Linear Regression Line

y = 0.6289x - 1.8759

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PM CEMS Response

PM Concentration

PS-11 Example 4C-R 19 Runs 10-27-04.xls Linear Summary 
CALCULATIONS FOR POLYNOMIAL CORRELATION

Facility: Facility D Emission Unit: BoilerLocation: Anytown USA Test Dates: 11/1-11/4/02, 1/7-1/8/03

PM CEMS PM concentration

response mg/acm CALCULATED STATISTICAL PARAMETERS

Run x y x2 x3 x4 xy x2y y^ (y^-y)2 delta 

1 6.7 3.1 45 301 2,015 21 139 2.906 0.038 0.2329 

3 7.8 2.1 61 475 3,702 16 128 3.246 1.313 0.1432 

4 9.2 3.7 85 779 7,164 34 313 3.770 0.005 0.1039 

5 10.1 6.1 102 1,030 10,406 62 622 4.161 3.761 0.1035 

6 11.1 2.8 123 1,368 15,181 31 345 4.644 3.400 0.1125 

7 12 4.9 144 1,728 20,736 59 706 5.123 0.050 0.1215 

8 18.2 8.3 331 6,029 109,720 151 2,749 9.569 1.610 0.0983 

9 13.7 4.5 188 2,571 35,228 62 845 6.144 2.702 0.1262 

10 14.6 8.2 213 3,112 45,437 120 1,748 6.745 2.118 0.1195 

11 19.7 13 388 7,645 150,614 256 5,045 10.944 4.227 0.1725 

12 18 11.1 324 5,832 104,976 200 3,596 9.394 2.910 0.0948 

13 20.3 11.9 412 8,365 169,818 242 4,904 11.527 0.139 0.2351 

14 20.2 10.6 408 8,242 166,497 214 4,325 11.428 0.686 0.2230 

15 19.7 9.2 388 7,645 150,614 181 3,570 10.944 3.042 0.1725 

16 5.2 2.4 27 141 731 12 65 2.543 0.020 0.4924 

17 8.6 4.8 74 636 5,470 41 355 3.533 1.606 0.1131 

18 14.1 6.2 199 2,803 39,525 87 1,233 6.406 0.042 0.1239 

19 14.5 7 210 3,049 44,205 102 1,472 6.676 0.105 0.1205 

20 17.3 8.6 299 5,178 89,575 149 2,574 8.800 0.040 0.0904 

PS-11 Example 4C-R 19 Runs 10-27-04.xls Polynomial Calcs
RESULTS OF POLYNOMIAL CORRELATION

Facility: Facility D Emission Unit: BoilerLocation: Anytown USA Test Dates: 11/1-11/4/02, 1/7-1/8/03

Variable Equation Value Correlation equation: y = 2.189 

n Number of data points = 19

S1 S1 = Sum (xi) = 261 Summary of Acceptance Criteria for PS-11

S2 S2 = Sum (xi2) = 4,021 Criterion Actual Allowable 

S3 S3 = Sum (xi3) = 66,929 Correlation coefficient 0.921 > 0.85 

S4 S4 = Sum (xi4) = 1,171,613 Confidence interval 4.54% < 10% 

S5 S5 = Sum (yi) = 129 Tolerance interval 12.1% < 25% S6 S6 = Sum (xiyi) = 2,039 * Indicates correlation coefficient is undefined.

S7 S7 = Sum (xi2yi) = 34,734

det A det A = nS2S4-S2S2S2+S1S3S2-S3S3n+S2S1S3-S4S1S1 = 5.945E+07 Check for Correlation Curve Minimum/Maximum

b0 b0 = (S5S2S4+S1S3S7+S2S6S3-S7S2S2-S3S3S5-S4S6S1)/det A = 2.189 Correlation curve minimum point 

b1 b1 = (nS6S4+S5S3S2+S2S1S7-S2S6S2-S7S3n-S4S1S5)/det A = -0.067 Minimum allowable x value 

b2 b2 = (nS2S7+S1S6S2+S5S1S3-S2S2S5-S3S6n-S7S1S1)/det A = 0.02596 Is correlation curve minimum < minimum x value? 

Sp Sp = Sqrt((1/(n-3)Sum of (y^-Y)2) = 1.318 Correlation curve maximum point 

D D = n(S2S4-S32)+S1(S3S2-S1S4)+S2(S1S3-S22) = 5.945E+07 Extrapolation limit for x (125% of maximum x value) 

C0 C0 = (S2S4-S32)/D = 3.902 Is correlation curve maximum > extrapolation limit? 

C1 C1 = (S3S2-S1S4)/D = -0.6164

C2 C2 = (S1S3-S22)/D = 2.182E-02

C3 C3 = (nS4-S22)/D = 1.024E-01C4 C4 = (S1S2-nS3)/D = -3.736E-03

C5 C5 = (nS2-S12)/D = 1.394E-04

tf tf from table = 2.120

EL Emission Limit = 18.5

CI CI = tf*Sp *sqrt(delta-min) = 0.840

CI% CI% = CI/EL*100= 4.54%

vf vf from table = 1.418

un' un' from table = 1.195

n' n' = 1/delta = 11.06

kT kt = un'*vf = 1.694

TI TI = kt*sp = 2.233

TI% TI% = TI/EL *100% = 12.1%

y~ y~ = 1/n*(Sum of (yi)) = 6.76

Sy Sy = sqrt(Sum of ( yi-y~)^2/(n-1) = 3.38

r2 r2 = 1-(Sp^2/Sy^2) = 0.847

r r = sqrt((1-Sp^2/Sy^2)) = 0.921

Max-min? b2 > 0? minimum 

xmax-min y = -b1/2b2 = 1.29 

1.25xmax 25.38 

* Indicates correlation coefficient is undefined.

Plot of Polynomial Regression C

y = 0.026x2 - 0.067x + 2.189

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PM Concentration

PS-11 Example 4C-R 19 Runs 10-27-04.xls Polynomial Summary 
CALCULATIONS FOR LOGARITHMIC CORRELATION

Facility: Facility D Emission Unit: BoilerLocation: Anytown USA Test Dates: 11/1-11/4/02, 1/7-1/8/

PM CEMS response PM concentration

Measured Transformed mg/acm CALCULATED STATISTICAL PARA

Run x x' = ln (x) y (xi'-x'~)2 (yi-y~)2 (x'i-x'~)(yi-y~) y^ 

1 6.7 1.902 3.1 0.418 13.419 2.367 23 7.8 2.054 2.1 0.244 21.745 2.305 34 9.2 2.219 3.7 0.108 9.383 1.008 45 10.1 2.313 6.1 0.056 0.440 0.156 56 11.1 2.407 2.8 0.020 15.707 0.560 57 12 2.485 4.9 0.004 3.471 0.118 68 18.2 2.901 8.3 0.125 2.362 0.543 99 13.7 2.617 4.5 0.005 5.122 -0.156 7

10 14.6 2.681 8.2 0.018 2.065 0.191 711 19.7 2.981 13 0.187 38.898 2.696 912 18 2.890 11.1 0.117 18.808 1.483 913 20.3 3.011 11.9 0.214 26.387 2.375 1014 20.2 3.006 10.6 0.209 14.721 1.755 1015 19.7 2.981 9.2 0.187 5.938 1.053 916 5.2 1.649 2.4 0.809 19.037 3.926 017 8.6 2.152 4.8 0.157 3.854 0.779 318 14.1 2.646 6.2 0.010 0.317 -0.055 719 14.5 2.674 7 0.016 0.056 0.030 720 17.3 2.851 8.6 0.091 3.374 0.555 8

PS-11 Example 4C-R 19 Runs 10-27-04.xls Logarithmic Calcs
RESULTS OF LOGARITHMIC CORRELATION

Facility: Facility D Emission Unit: BoilerLocation: Anytown USA Test Dates: 11/1-11/4/

Correlation equation: y = -11.697 

Variable Equation Value

n Number of data points = 19 Summary of Acceptance Criteria for PS-11

x'~ x'~ = 1/n*(Sum of (x'i)) = 2.55 Criterion Actual Allo

Sx'x' Sx'x' = Sum((x'i-x'~)^2) = 2.99 Correlation coefficient 0.867 > 0

y

~ y~ = 1/n*(Sum of (yi)) = 6.8 Confidence interval 4.40% < 

Syy Syy = Sum((yi-y~)^2) = 205 Tolerance interval 15.0% < 2

Sx'y Sx'y = Sum((x'i-x'~)(yi-y~)) = 21.7 * Indicates correlation coefficient is undefined.

b0 b0 = y~-b1x'~ = -11.697

b1 b1 = Sx'y/Sx'x' = 7.244

SL SL = SQRT(1/(n-2)(Sum(yi^-yi)^2)) = 1.680

y^mean y^ at mean x value = 6.763

tf tf from table = 2.110

CI CI = tf*SL*SQRT(1/n) = 0.813

EL Emission Limit = 18.5

CI% CI% =CI/EL*100% = 4.40%

n' n' = n = 19

vf vf from table = 1.400

un' un' from table = 1.178

kT kt = un'*vf = 1.649

TI TI = kt*SL = 2.77

TI% TI% = TI/EL*100% = 15.0%

Sy Sy = SQRT(Syy/(n-1)) = 3.376 

r2 r2 = 1-(SL^2/Sy^2) = 0.752 

r r = SQRT((1-SL^2/Sy^2)) = 0.867 

* Indicates correlation coefficient is undefined.

Plot of Logarithmic Regression C

y = 7.244Ln(x) - 11.697

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PM CEMS Response

PM Concentration

PS-11 Example 4C-R 19 Runs 10-27-04.xls Logarithmic Summary 
CALCULATIONS FOR EXPONENTIAL CORRELATION

Facility: Facility D Emission Unit: BoilerLocation: Anytown USA Test Dates: 11/1-11/4/02, 1/7

PM concentration

PM CEMS mg/acm

response Measured Transformed CALCULATED STATISTICAL PARAMETERS

Run x y y' = Ln(y) (xi-x~)2 (y'i-y'~)2 (xi-x~)(y'i-y'~) y'^ (y'

1 6.7 3.1 1.131 49.52 0.414 4.530 1.042 3 7.8 2.1 0.742 35.25 1.068 6.134 1.156 4 9.2 3.7 1.308 20.58 0.218 2.118 1.302 5 10.1 6.1 1.808 13.23 0.001 -0.120 1.396 6 11.1 2.8 1.030 6.95 0.556 1.966 1.500 7 12 4.9 1.589 3.02 0.035 0.323 1.594 8 18.2 8.3 2.116 19.92 0.116 1.522 2.240 9 13.7 4.5 1.504 0.00 0.073 0.010 1.771 

10 14.6 8.2 2.104 0.75 0.108 0.284 1.865 11 19.7 13 2.565 35.56 0.624 4.710 2.397 12 18 11.1 2.407 18.17 0.399 2.693 2.219 13 20.3 11.9 2.477 43.08 0.492 4.603 2.459 14 20.2 10.6 2.361 41.77 0.343 3.785 2.449 15 19.7 9.2 2.219 35.56 0.197 2.648 2.397 16 5.2 2.4 0.875 72.88 0.809 7.681 0.886 17 8.6 4.8 1.569 26.39 0.043 1.061 1.240 18 14.1 6.2 1.825 0.13 0.002 0.018 1.813 19 14.5 7 1.946 0.58 0.029 0.130 1.855 20 17.3 8.6 2.152 12.70 0.142 1.342 2.146 

PS-11 Example 4C-R 19 Runs 10-27-04.xls Exponential Calcs
RESULTS OF EXPONENTIAL CORRELATION

Facility: Facility D Emission Unit: BoilerLocation: Anytown USA Test Dates: 11/1-11/4/02, 1/7-1/8/03

Correlation equation: y = 1.410 e 0.1

Variable Equation Value

n Number of data points = 19 Summary of Acceptance Criteria for PS-11

x

~ x~ = 1/n*(Sum of (xi)) = 13.74 Criterion Actual Allowable 

Sxx Sxx = Sum((xi-x~)^2) = 436.0 Correlation coefficient 0.909 > 0.85 

y'~ y'~ = 1/n*(Sum of (y'i)) = 1.775 Confidence interval 3.63% < 10% 

Sy'y' Sy'y' = Sum((y'i-y'~)^2) = 5.67 Tolerance interval 12.6% < 25% 

Sxy' Sxy' = Sum((xi-x~)(y'i-y'~)) = 45.44 * Indicates correlation coefficient is undefined.

b0' b0' = y'~-b1x~ = 0.344

b0 b0 = e^b0' = 1.410

b1 b1 = Sxy'/Sxx = 0.104

SL SL = SQRT(1/(n-2)(Sum(y'i^-y'i)^2)) = 0.235

y'^mean y'^ at mean x value = 1.775

tf tf from table = 2.110

CI' CI' = tf*SL*SQRT(1/n) = 0.1135

LCL' LCL' = y~' - CI' 1.662

UCL' UCL' = y~' + CI' 1.889

CI CI = (e^UCL' - e^LCL')/2 0.671

EL Emission Limit = 18.5

CI% CI% =CI/EL*100% = 3.63%

n' n' = n = 19

vf vf from table = 1.400

un' un' from table = 1.178

kT kT = un'*vf = 1.649

TI' TI' = kt*SL = 0.387

LTL' LTL' = y~' - TI' 1.388

UTL' UTL' = y~' + TI' 2.162

TI TI = (e^UTL' - e^LTL')/2 2.340

TI% TI% = TI/EL*100% = 12.6%

Sy' Sy' = SQRT(Sy'y'/(n-1)) = 0.561 PS-11 Correlation 

r2 r2 = 1-(SL^2/Sy'^2) = 0.825 Calculation Spread

r r = SQRT((1-SL^2/Sy'^2)) = 0.909 Version 2-6 10/2

* Indicates correlation coefficient is undefined.

Plot of Exponential Regression Curve

y = 1.4101e0.1042x

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PM CEMS Response

PM Concentration

PS-11 Example 4C-R 19 Runs 10-27-04.xls Exponential Summary 
CALCULATIONS FOR POWER CORRELATION

Facility: Facility D Emission Unit: BoilerLocation: Anytown USA Test Dates: 11/1-11/4/02, 1/7-

PM concentration

PM CEMS response mg/acm

Measured Transformed Measured Transformed CALCULATED STATISTICAL PARA

Run x x' = Ln(x) y y' = Ln(y) (x'i-x'~)2 (y'i-y'~)2 (x'i-x'~)(y'i-y'~) 

1 6.7 1.902 3.1 1.131 0.418 0.414 0.416 3 7.8 2.054 2.1 0.742 0.244 1.068 0.511 4 9.2 2.219 3.7 1.308 0.108 0.218 0.154 5 10.1 2.313 6.1 1.808 0.056 0.001 -0.008 6 11.1 2.407 2.8 1.030 0.020 0.556 0.105 7 12 2.485 4.9 1.589 0.004 0.035 0.012 8 18.2 2.901 8.3 2.116 0.125 0.116 0.120 9 13.7 2.617 4.5 1.504 0.005 0.073 -0.019 

10 14.6 2.681 8.2 2.104 0.018 0.108 0.044 11 19.7 2.981 13 2.565 0.187 0.624 0.341 12 18 2.890 11.1 2.407 0.117 0.399 0.216 13 20.3 3.011 11.9 2.477 0.214 0.492 0.324 14 20.2 3.006 10.6 2.361 0.209 0.343 0.268 15 19.7 2.981 9.2 2.219 0.187 0.197 0.192 16 5.2 1.649 2.4 0.875 0.809 0.809 0.809 17 8.6 2.152 4.8 1.569 0.157 0.043 0.082 18 14.1 2.646 6.2 1.825 0.010 0.002 0.005 19 14.5 2.674 7 1.946 0.016 0.029 0.021 20 17.3 2.851 8.6 2.152 0.091 0.142 0.114 

PS-11 Example 4C-R 19 Runs 10-27-04.xls Power Calcs
RESULTS OF POWER CORRELATION

Facility: Facility D Emission Unit: BoilerLocation: Anytown USA Test Dates: 11/1-11/4/02, 1/7-1/8/03

Correlation equation: y = 0.2513 x 1.2385

Variable Equation Value

n Number of data points = 19 Summary of Acceptance Criteria for PS-11

x'~ x'~ = 1/n*(Sum of (x'i)) = 2.548 Criterion Actual Allowable 

Sx'x' Sx'x' = Sum((x'i-x'~)^2) = 2.99 Correlation coefficient 0.894 > 0.85 

y'~ y'~ = 1/n*(Sum of (y'i)) = 1.775 Confidence interval 3.90% < 10% 

Sy'y' Sy'y' = Sum((y'i-y'~)^2) = 5.67 Tolerance interval 13.6% < 25% Sx'y' Sx'y' = Sum((x'i-x'~)(y'i-y'~)) = 3.71 * Indicates correlation coefficient is undefined.

b0' b0' = y'~-b1x~ = -1.381

b0 b0 = e^b0' = 0.251

b1 b1 = Sx'y'/Sx'x' = 1.239

SL SL = SQRT(1/(n-2)(Sum(y'i^-y'i)^2)) = 0.252

y'^mean y'^ at mean x value = 1.775

tf tf from table = 2.110

CI' CI = tf*SL*SQRT(1/n) = 0.122

LCL' LCL' = y~' - CI' 1.653

UCL' UCL' = y~' + CI' 1.897

CI CI = (e^UCL' - e^LCL')/2 0.721

EL Emission Limit = 18.5

CI% CI% =CI/EL*100% = 3.90%

n' n' = n = 19

vf vf from table = 1.400

un' un' from table = 1.178

kT kT = un'*vf = 1.649

TI' TI' = kTSL = 0.415

LTL' LTL' = y~' - TI' 1.360

UTL' UTL' = y~' + TI' 2.190

TI TI = (e^UTL' - e^LTL')/2 2.521

TI% TI% = TI/EL*100% = 13.6%

Sy' Sy' = SQRT(Sy'y'/(n-1)) = 0.561 PS-11 

r2 r2 = 1-(SL^2/Sy'^2) = 0.799 Calcula

r r = SQRT((1-SL^2/Sy'^2)) = 0.894 Version 

* Indicates correlation coefficient is undefined.

Plot of Power Function Regression Curve

y = 0.2513x1.2385

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PM CEMS Response

PM Concentration

PS-11 Example 4C-R 19 Runs 10-27-04.xls Power Summary 
SELECTION OF BEST MODEL

Facility: Facility D Emission Unit: BoilerLocation: Anytown USA Test Dates: 11/1-11/4/02, 1

Model Correlation coefficient (a) > 0.85 ? Confidence interval half range percentage < 10% ? Tolerance interval half range percentage < 25% ? Min/max wit

allowable ran

Linear 0.912 Yes 3.63% Yes 12.4% Yes (b) 

Polynomial 0.921 Yes 4.54% Yes 12.1% Yes Yes 

Logarithmic 0.867 Yes 4.40% Yes 15.0% Yes (b) 

Exponential 0.909 Yes 3.63% Yes 12.6% Yes (b) 

Power 0.894 Yes 3.90% Yes 13.6% Yes (b) 

(a) * indicates correlation coefficient is undefined; model does not satisfy criterion.(b) Not applicable; criterion applies only to polynomial model.

Best m

PS-11 Correlation TestCalculation SpreadsheetVersion 2-6 10/25/04

PS-11 Example 4C-R 19 Runs 10-27-04.xls Best Model